I'll be getting into the way they act and a couple theories I have.
As mentioned in the last entry Bruce has shown that he trusts Jason. He chose him for a mission to decide the fate of the world (Batman/Superman Annual #1), picked him as Wingman (saying he trusted him in Batman INC.), told Superman he vouched for him and calls him in for assistance several times. In Batman Eternal Jasons' the one Bruce asks to watch after Batgirl. He even admits Jason has experience controlling his anger showing he has seen Jason making efforts to heal himself.
Jason himself has been shown trying to be forgiving. Not just to move on for his own peace of mind but because he actually cares. Surprisingly he's more forgiving than the other bat kids in some cases. He's tried to reach out when Bruce was upset and almost gave a last message when he thought he was going to die again (Batman Eternal.) It's a struggle since Bruce doesn't always come off so well since he deceives and isn't always willing to admit when he's in the wrong.
One of the biggest strains on their relationship is naturally Jasons' death but let's discuss the less talked about aspect. Back in Under the Hood Jason gave Batman blood, tissue and fingerprints to prove his identity. Although part of Bruce suspected Jasons' return he wouldn't accept it without evidence. As a detective he needs to figure out the truth before he can fully believe something. Jason himself points this out which was his reason for handing over the proof. Prior to Red Hoods' identity being revealed Batman visits several people who either came back from the dead or have insight into resurrections. All because he has a hunch Jason has returned but can't bring himself to admit it. But he never finds out how his adopted son came back and Jason doesn't know or care. (*1)
We don't know how this played out in the new continuity but it begs the question: can Bruce really accept something fully without knowing all the answers? The worlds' greatest detective doesn't know how one of his kids came back from the dead. In Batman INC. Alfred seems to think Talia was entirely responsible. Why is unclear as her involvement was never known to the bat family prior to this. Tynion seemed to imply it was Talia that resurrected him via the Lazarus pit as well (for vague reasons that don't make sense. *2) Under Lobdell it was stated a handful of times that Jasons' return was a mystery. (In fact we see him dig himself out of his grave in Secret Origins.) Tomasi uses this angle for Bruces' confrontation with Jason as he tried to jog something from his memory.
Was Bruce trying to figure out the truth on his own ever since he proved this was indeed Jason Todd? It does seem fairly ridiculous to assume he'd shrug off the whole matter. Bruce is a driven individual who had a family member return from the great beyond and was obsessed with death for most of his life. I think he would likely hit many dead ends while trying to figure out the truth and being unable to solve the mystery might be a big factor for him not entirely accepting Jason.
In Justice League #19 by Geoff Johns we see Alfred putting up a case for Damian while discussing it with Jason. The exchange does give some insight on their thought processes. Alfred for some reason doesn't seem to believe Jason died. (*3) Maybe he's simply in denial about a world he doesn't want to think about. Jason doesn't think his death mattered because he wasn't a "real" son, just a sidekick. Alfred chides this but really why would Jason think he meant more? Even back in RHATO #6 he thinks that Bruce pretended to love him like a son and he's had little proof that he mattered since then. In fact some of the behavior he sees makes it seem like he's still dead. Damians' case is destroyed and he never gets a new one. Yet Jason always gets a replacement and even though he's alive he still has a case in place as if he never came back.
Bruce also personally destroyed Damians' headstone and for all we know Jasons' still exists. He wasn't included in the family portrait either. And when Damian assumes Jasons' case was meant to honor him Bruce claims it wasn't. That it's supposed to act as a reminder of what happened. As if Jason himself wasn't important but Batmans' failure was. I'd like to think this behavior is simply because Bruce has trouble believing Jason has truly returned until he knows all the facts. What if Jason isn't back for good or his mission get him killed again? Bruce has taken Damians' resurrection in strides although the death lasted for a shorter period (from his POV) and he knows how Damian was brought back. Bruce was totally sideswiped by Jasons' return and continues to be left in the dark.
Curiously he hasn't said anything about the bat symbol Jason wears which shows that despite his choices Bruce wants him in the fold. Something that's done at a distance giving both some much needed space. When Arsenal suggests Jason and him become legitimate crime fighters that get paid Jason has this thought:
Jason Todd: "At heart, I'm a loner. Roy? He's always wanted to be part of something bigger. With Ollie. Then the Outlaws. Now this. I used to live with Batman. Batman. I've practically got a PHD. in how to screw up in an epic way."
Lobdell has noted that Jason is conflicted between wanting to be a loner and a desire to be around others. This statement does suggest he's willing to go along with these plans because Roy wants it so badly but there's more to it. The bit about Batman could mean his past with the Dark Knight makes him unsure of himself. It could also mean that he's afraid Batman taught him too many of his anti-social interactions to the point he's not sure if he can change himself enough to be part of something bigger.
It has been noted that Jason is an unusual case in the family as Bruce has loosened up on the no kill rule and apparently accepts that Jason will take lives. Except when he's in Gotham or missions he takes with other bats. This was brought up after Jason mentions the fact Bruce vouched for him.
Bruce Wayne: "I'm never going to approve of your methods, Jason. But I can't argue with your results. I learned a long time ago I can't control everyone. Nor should I try."
Jason Todd: "Really."
Bruce Wayne: "You deserve it."
This does show that Batman sees the good Jason does despite not being a fan of the lethal methods. Still his statement about knowing he can't control everyone is mostly BS. as we have and will continue to see him manipulate others, the bat family included. Naturally Jason expresses doubt and Bruces' answer does seem to imply he's making an exception. I don't think this is out of guilt but rather something he has given a lot of thought. We later see Bruce recall being impressed with Jason as Robin for pointing out he needed to learn from his own mistakes. Perhaps he finally realized any change has to come from Jason. It also might have occurred to him that everyone deals with their traumas in different ways and he can hardly blame Jason for the way he sees things.
Plus the meeting took place not long after DOTF where Bruce nearly lost everyone. Most of the family were mad at him. It only seems natural that he'd want to make an attempt at burying the hatchet. There's another bit I think is overlooked by many and that's the fact Bruce has given another family member similar leeway. Alfred Pennyworth. Prior to this Alfred was the only family member he's ever allowed to carry a gun. The butler has never shied away from using one and seems more than ready to kill if he need be.
What I find more questionable is if Bruce is really willing to use Jason to kill enemies. This was brought up in the Batman/Superman Annual #1 when he says it's one of the reasons he chose Jason for the mission. Would he have also picked Damian to kill for him if he had been around? It seems like a disservice to Jason if he wouldn't. We never actually see if anyone was killed by Jason although Bruce does debate the issue throughout the story. One of the great ironies being when Bruce asks if he really believes people can't change. Jason points out that they keep having the same argument. Earlier in the book Bruce himself stated Jason never changes. Which isn't true in the slightest, not only because of his emotion growth but the fact Jason shows more restraint than he previous had as Red Hood. Out of the two of them Bruce is the one that often struggles with change. He also knows some people can't change their nature, like the Joker.
Why Bruce doesn't kill has changed from creator to creator. In UTH he seems to believe taking a life would take him down a path where he'd be unable to stop killing. Snyder seems to suggest that Bruce is somewhat superstitious in thinking that killing the Joker would bring about a greater evil. The only way I could see the two being connected is if Bruce fears he'll be worse than Joker if he crossed that line. Which suggests some distrurbing things about the dark knight. Still if this was his chief concern, not just trying to be morally superior, it makes the subject of their friction even more tragic. Jason believes by not killing Joker Bruce doesn't care about him. What if Bruce feared crossing the line would lead to him becoming like the clown prince then harming those he cared about?
*1 Somehow pre-Flashpoint Jason was able to figure out the basics of his resurrection during Countdown.
*2 We never get any information why Bruce doesn't believe the pit brought Jason back. Tynion claims Tomasi and him talked over the phone to ensure their stories flowed right. Yet Tynion implies the pit brought Jason back and Tomasi has Bruce exposing Jason to trauma because he believes otherwise. Since neither story did anything with the plot Tomasi set up it's all pretty moot. Also for some reason Tynion says it's always been Talias' plan to make Jason the heir to the League of Assassins despite many previous stories suggesting otherwise.
*3 Correction: Alfred wasn't referring to Jason's death in DITF but the fact he almost died from the acid helmet Joker set up. I guess it just shocks me when writers actually acknowledge RHATO since so many ignore It.