Since pre-52 often portrayed Jason as a different person appearance to appearance/or had things constantly retconned, this will mainly focus on the new 52 characterizations. To get some insight on one of the most complex relationships in the bat family.
Let's start off from the beginning when the two met according to new canon of DCnU. RHATO #0 is a bit vague on the specifics but some of the points remain the same. Batman finds Jason in an alleyway stealing then decides to help him get a better life. In this version he takes the boy into his home before deciding to make him Robin. While much is kept out of this origin it is clear that he has little to no interest in Bruce Wayne and feels more of a connection with Alfred, at least at the start. Unlike the other Robins Jason doesn't seek the role but rather Bruce (and Alfred in this case) ask him to be Batmans' partner. (*1)
What went wrong varies usually based on the writer/character speaking. In the relaunch I believe Lobdell has been the only one to directly address this although Tomasi had Damian victim blaming Jason. Lobdell has both Bruce and Jason acknowledge there being problems with the second Robins' anger. Jason admits it had something to do with fighting the demons of his past while trying too hard to prove himself to his new father figure. While it hasn't been explored in depth in this canon I think the problem of Bruces' issues with dealing with emotions (rather not dealing with them) were also a big issue. This however IS obvious in a later interactions between the two as well as the fact Bruce isn't always honest. The latter mostly being something Lobdell (and Tomasi) have shown although Jason figured out Bruce was using him to help himself heal. When confronted Bruce doesn't deny the accusation.
Not feeling that he belongs Jason goes to Africa to find his mom, dies but doesn't blame Bruce for his death. Once he returns he's naturally upset because Bruce hasn't avenged him which to him means that his adopted father never cared about him. Which isn't a surprising suspicion since Bruce is often horrible at conveying his feelings. During Jasons' life there weren't many tells and after his death there's no obvious changes that he meant anything. Life went on as if he never existed with only his grave to mark his passing and another kid filling the sidekick spot. Jason might not have known about the case (until much later) but even if he did it won't prove anything. Some versions have a plaque with "good solider" written on it which doesn't give the impression of family. It's detached to the point it looks like Jason Todd was just another solider in a war not a son. At least one version has "friend and ally" on his headstone which also gives a distant feeling.
In UTH Jason tries to get a reaction from Bruce, some proof that he mattered. Unlike Dick or Tim he never had a happy family life before. Bruce was the only semi-decent parental figure he had and only family he really felt he had left. There's literally nothing left in his life but what he had before he died. Nothing worth fighting for (*2) or living for other than the showdown between the Joker, Bruce and him. There's a very real death wish throughout this because deep down he already knows the answer to the question he's forcing out of Bruce. Would he being willing to save Jason no matter the cost? Does he matter more than the no kill rule? He knows Bruce won't chose him and is ready to die because he has nothing else. But Bruce not only saves Jason's murderer he slices Jason's neck open in the process.
During UTH there were moments that highlight how alike the two were by what they did and how they reacted in similar manners. This wasn't just to say Jason knew how Batman thought or to show the training Bruce gave him. At one point Black Mask realizes they know each other because they both say the same exact thing about him. A dismissive remark that actually hints at them seeing more eye to eye than you'd expect. Before the final showdown they even fight together for the first time since DITF and it's natural for both of them. The relaunch has both of them cool down a lot since this storyline. Some of Jason's journey is shown but some of it hasn't been fleshed out quite yet.
Now it's hard to mesh Morrsions' Batman INC. with the rest of the new 52 stories. At the start of Batwoman there's a warning Batman gives her about sidekicks dying and/or turning into villains. This was a subject of debate as to the identity of Batman given the length of time it took to get the Batwoman series started. It's certainly a harsh comment something that at the very least seems odd for him to say to Kate since they aren't close. Yes it's to get her to look out for Bette Kane but it also has that familiar victim blaming vibe to it which often seems like Bruce is trying to separate himself from the equation. Both emotionally and from the blame. It suggests to me that yes Bette reminds him of Jason but also that he's frustrated/angry with Jason when this takes place.
I can't help reading the exchange and seeing it taking place months before the new 52 starts. Sometime after Jason has returned as it reflects the same hurt Winick wrote in Batmans' words to Ollie on the subject in his pre-Flashpoint Green Arrow run. There Bruce was worse referring to Jason as more trash in the sewers. Ollie saw past the comment there but ironically they both blamed each other for what happened with Jason and Roy. (*3) Bruce tends to get nasty when he's upset which has been the case for other bat kids too. At the start of RHATO Jason is dismissive of Batman and the rest of his allies. Acting as if he doesn't care what happens to them and that he believes their worse than the villains they fight. Roy readily buys this as he assumes Jason won't go to their aid and Jason admits to himself later on that he doubts they'd lift a finger for him.
It's not until issue #3 we see how much the fatherly relationship he had with Bruce meant to him. The most cherished memory he had was Bruce giving up the mission for the night to spent time with him when Jason was sick. It meant everything for several reasons: No one ever took care of him before--HE was the caretaker of his parents and usually had to fend for himself. For Bruce to ignore the mission for the night was huge and it's likely the only time he did. Doing so means Jason matters and he's more than a Robin he's a son. Jason gives up the memory likely knowing it's importance but needing to sever his ties to the bat family because it hurts too much.
In NOTO Jason fully believes there's a chance Batman and his allies would set him up. That they'd throw him in jail if they got a chance. It's unknown if past canon of being in jail happened in this timeline. (*4) He's clearly conflicted and upset about being in the city during this time. When would Bruce approach him to ask for him to be Wingman? It'd have to be before DOTF. The only time that would work would be after fighting the Talons and before the Outlaws go into space. It has to be then because of Jason's uncertainty about helping out in #9 and right after the space arc is DOTF/Damian's death. Even then it's not a perfect fit but prior to all this Jason turns up for Damian's meeting simply because he believes Tim invited him.
During this time he exchanges no words with Bruce and shrugs off Damian's perception of his relationship with him. Jason also sarcastically remarks about the fact he wasn't part of the family portrait. Some have noted that it makes sense to exclude Jason since he's legally dead but no one seems aware of Jasons' identity despite him forgoing a mask around so many people. Even if they were on bad terms (unclear where this fits in canon) Bruce could ask the painter to add Jason in using a photo (Bruce would try to track him.) I'm sure he could find a painter who wouldn't blab if he was worried about secrecy, maybe even a friend.
The fight between Damian and the Robins could take place after Jason became Wingman but even that sounds cold. Just like Bruce bringing Jason up as an afterthought to mention that even Jason did well as Robin. At one point Damian claims his father said some negative things about Jason. Which could just be Damian being an ass and/or partly true. Pre-Flashpoint Batman did victim blame, so did most of the bat family which makes it hard for me to completely dismiss it all as simply Damians' uninformed opinion.
Since the Robins ended the meeting on a semi-good standing (who knows why given Damian's behavior) it's not too strange to see him with them in DOTF. Again it's hard to map how canon is supposed to flow with Morrisons' work. In Batman INC. Jason helps take down Talias' assassins and ends up saving Bruce's life. I'm honestly not sure if Bruce was really clueless to Jason duping Talia but he seems shocked and betrayed at the idea of Jason giving her the doomsday device even if it's in exchange for his life. Yet once he saves Bruce the two have nothing to say to each other. In fact Bruce acts like he was the reason Talia lost everything. Jason just stands there as his mentor and Talia rant at each other. He silently leaves just as Kathy Kane does, maybe because the ending is supposed to make Batman look foolish or a dozen other reasons. It is odd that Jason doesn't address Bruce even when he gives his condolences to Talia after hearing Damian died.
DOTF has Bruce dreaming of the bat family all being close. While it's assumed that most of them are on friendly terms the most noticeable change is Jason's inclusion. Bruce wants Jason there so he's part of the group. He's pretty much the focus as Damian snarks at him and Dick gives Jason credit. What did he do to earn such praise? Caught the Joker instead of killing him, doing what Bruce wants him to do. Not just obeying his rules but letting go of the past. When Alfred turns into Joker while the others are backing away from the attack Jason is seen protecting Tim. Bruce knows Jason died trying to protect his mother from the explosion and likely knows these two have grown closer. In his mind Bruce knows it's something Jason would naturally do.
Bruce wakes up to find the family displeased with Jason reacting surprisingly tame about the Joker threatening all of them. (It should be noted that he raises some good points that actually turn out to be true.) He tells it like it is yet he doesn't quite hit Bruce's buttons as well as he usually does. It's mostly the kids minus Damian saying their piece. In his own title Jason wonders if Jokers' tale of creating him as Robin is true meaning his life until his death was all planned. He returns to the manor to say goodbye to the others ready to leave without seeing Bruce who of course knows what he was intending.
Jason mentions his encounter with Superman who claimed (to Jason's utter shock) that Batman vouched for him. Bruce says that he'll never condone the actions although he can't argue with the results. In a moment of vulnerability Jason admits he's unsure about what Joker said about creating him. Interestingly Bruce doesn't seem shocked by this theory but assures Jason no matter what happened neither he nor Joker made him who he is. Touched Jason leaves the cave unable to voice how grateful he is to hear that. When Jokers' trap is sprung Bruce hears the scream from the cave and is somehow the first to arrive at the scene.
Instead of going on patrol he stays by Jason's bedside feeling guilty for what happened. While Jason is in an induced coma Bruce says the reason for him never having to worry about his adopted son. When Jason was training he accepted blame for his own mistakes and said it would help him learn. After he wakes Jason starts to apologize only to be stopped by Bruce who hugs him as the two finally forgive each other. Apparently Jason stayed at the manor until he healed sometime after Damian's death. (How that works is unclear but just roll with it.) Going through his jerk phrase of the healing process Bruce started alienating most of his family. What he does to Jason is in my opinion the worst thing he's ever done to any of them. This wasn't just a spur of the moment decision, Bruce planned this whole trip without his adopted sons' consent to take him to the place he died. Why? Just to trigger his PTSD in in hopes it would jog Jason's memories of how he returned from the dead.
The moment completely undermines the progress they made and is quickly swept away instead of dealing with the issues the story created. Other than a brief mention in Tynions' RHATO run this hasn't been brought up in any other title besides Batman and Robin. Interestingly the first Batman/Superman annual ignored this completely and introduced a story that required the two to work together in a situation that might require killing. Asked to bring in two members of their family to fight for the fate of their world Batman only has one choice: Jason. The two do have a somewhat tense relationship in the annual. When it occurs to Bruce that there could be more death and he recalls all the loss in their lives. Jason tells him off for acting like it's any different than all the other times they risked their lives. Superman notes how much his words hit their target by listening to Bruces' heart rate rise. Later when asked if their family Jason doesn't deny it just state their not related by blood.
There have been some subtle hints at other problems in their relationship in RHATO. In #9 Jason gets annoyed when Freeze says he's not some idiot that needs someone to save him. While this does connect with Jason trying to push Kori and Roy away the issue does bring back his past. How being in his city again makes him feel, his relationship with the bat family, his death and whether he's actually welcomed there. Originally the issue was going to end with him throwing his captured enemy at Bruces' feet and retorting something along the lines of , "Don't say I never do anything for you." This was nixed because of the timeline set for Batman during this event. Still Freezes' words piss Jason off suggesting his irritation is deeper than just his misgivings with working with his friends. Perhaps lingering frustration with being unable to save himself in DITF? Maybe seeing working with others as proof of some perceived weakness?
Interesting Jasons' interaction with Oliver Queen under Lobdell gives us another less explored piece of the puzzle. Ollie tries to prove he can do more for Roy than his friends by making a call. Jason comments on this being the standard way rich guys react by thinking that money solves everything. Does this mean Bruces' charity work or something to do with his parenting skills? Did Bruce believe he could smooth things over with presents while remaining emotionally aloof ? It seems like a possiblity and if nothing else shows that as a poor kid Jason would have a different view of the way Bruce acts. We definately know the dark knight has trouble opening up and communication was one of the problems in their relationship before the relaunch. Although he seems to be this way with everyone as Bruce promised no more secrets while neglecting to mention Dick Grayson was alive. While it could be argued he tries to protect them (like leaving out the fact Jokers' card ended up in the cave) he has been manipulative as he was during his grieving process for Damian. In the case of DOTF he still didn't tell them that he revealed his face to Joker.
I did find it interesting during their fight in Batman and Red Hood that Jason points out a key difference between them. The same thing he mentions in Secret Origins, that Jason tries to let go of the past while Bruce clings to it. That's why no matter how many times Bruce keeps making the same mistakes he's doomed to repeat them. That means Bruce is locked in the same cycle that was self destructive for Jason because he'll always be obsessed with the same things. Lobdell has stated in an interview that Jason also allows himself to take time off and realizes he has to do manageable missions. Batman on the other hand is fighting a battle he's doomed to lose. We've seen it in so many possible futures just like we've seen Bruce alone because he simply can't let go. He hasn't and will never learn what Jason has. I'm not saying Jason is completely over what happened to him, just like in real life I doubt that tramua will ever go away. But the fact remains he is dealing with it in a healthier manner.
Look at his interactions with people (when written well) compared to Bruce. Thus far none of the family have done anything for him that doesn't have to do with a missions their involved in. No one has come to his aid (unless their already on a shared mission) yet he always answers their request for assistance. Jason has even played by their rules. He's taken what he's learned from the All Caste and his friends. There are story arcs for Bruce as well but their usually forgotten soon after.
*1 Batman also offered Jason the role post-crisis, in that case before he even revealed his ID. There has been hints of this happening in other versions of the origin too.
*2 Lobdell has introduced the All Caste Jason's teachers upon his return to life which he later confesses being perhaps the best people he's met. It seems like he didn't realize this until they died as he was too consumed by his anger to consider their lessons and meaning beforehand. Even so with the exception of Ducra none of them seem close enough to be considered family. When they kicked Essence out he left feeling they were in the wrong and assuming they abandoned her.
*3 In yet another mention to Jason and Roy when Ollie was in heaven (pre-Flashpoint) he saw Jason. Apparently Jason was said to be close to Roy and Ollie assumed Roy got him hooked on drugs and that was what killed him. Not much makes sense about the exchange regardless since Ollie died sometime after Jason and would hear about the death. Not to mention Jason would be alive by this point even though it's now been confirmed that he was in heaven at some point.
*4 In RHATO Jason mentions previously spending time in Arkham and hating it. The reason is never clarified so we don't know if his family put him there or he went undercover like Dick did at the start of New 52. If it was the former his family are assholes. I didn't buy the reasons Winick had Batman give because it was bullshit.