The journey into becoming someone else and the psychology of what they do. Some other thoughts that came to mind too.
One could argue that all of the bat kids took heroes journeys to become someone else. But Judd Winick purposely made Jasons' post-rebirth training similar to Bruces'. Both had a need to recreate themselves due to tragedy and sought out people that could train them. The relaunch had to change that given the shortened time period by having the All Caste train him. (*1) No one else in the family took a trip to find other teachers to better themselves in secret. (*2) In Zero Year we learn that Bruce was declared legally dead during this period. His uncle Phillip forces him to come out of hiding to try to make him take on the family responsibilities. While it might be somewhat self motivated (*3) Bruce never makes Jason publicly return.
In UTH he mentions the fact that some criminals don't fear Batman because they know he doesn't kill. Jason would know since he grew up in an environment filled with crime. In a lot of ways Jason is more practical than Bruce and he's also more detached. Gotham may be his home but he's distanced himself to the point he's not obsessed with it. He has no particular city to protect, no where that he's lived for more than a short period. Even the island he shared with Kori and Roy often acted as a pit stop before the next adventure. Batman sees his costume as something to strike fear while Jason dislikes the whole "dress up" part of crime fighting. His uniform reflects that with armor, weapons and a causal jacket/hoodie to complete the look.
The bat suit was created to scare his foes while Jasons' Red Hood uniforms make their own statement. At first it's to get the attention of Batman and Joker although there is an element of mockery to it. Winick admitted he never thought further into the name than Jason using it to get to those two. Still what the name represented is important. The Red Hood gang was basically filled with replaceable people. In the Killing Joke it was the identity used to bridge who Joker was and who he would become after his rebirth. Jason views himself as nothing more than a placeholder for Robin (JL #19) and is a different person after his resurrection. I never brought Morrisons' version of the Red Hood look since it lacked Jasons' style (along with other reasons.) He might be mocking others but at least with his original outfit and the ones he wears in his own series he still retains a sense of self.
As pointed out in RHATO the bat symbol he wears is a statement piece that most are likely unaware of. It actually means he's not going to let himself be defined by his past. This might have changed somewhat since he patched things up with Bruce and took another bat symbol. Interestingly he's the only non-bat themed character to wear one.
Technically Bruce has never died though Endgame does have death themes as he considers the likelihood of his own passing. His thoughts go back to the solitude he felt when he fell and believed he died. The comfort he found being in the dark by himself. This is eerily similar to Jasons' own thoughts while on Venom and recalling the peace he felt in his grave. Both yearn for it and being separated from others.
There has been some debate on how crazy Joker is. Real life psychiatrists have said that Joker wouldn't be considered legally insane, a point that's brought up in comics from time to time. Still in the past Jokers' insanity is the only reason he avoids death and one would assume Batman would prove he wasn't to make sure justice prevails. Snyder has his Batman flat out say Joker isn't insane but rather evil. Then again Batman was wrong about Joker not knowing his identity under Snyder. I always felt the people that would know best are those Joker tormented. That's why Winicks' view in UTH rings true to me. Jason knows Joker is crazy just not as insane as he's pretending to be. Jokers' not pleased to hear this implying it's true.
Jason admits in RHA #4 that he always held back when fighting Bruce which fits perfectly with what Winick wrote. The whole point of Jason Todd' return story wasn't him trying to kill Batman or even proving himself superior. It was about making Bruce chose to see what was more important to him. Despite Jason not going all out against his mentor Bruce still has a moment where he seriously worries Jason has surpassed him. Yes, he's that good and Dick also sees the skill to the point he's impressed too. Most writers haven't noticed this often acting like he isn't as good when in reality he's someone Batman concerned about even though he wasn't giving it his all.
*1 The timeframe is one of the reasons the later retcon of the League of Assassins makes no sense. It's said that Jason needed them to train him to be an assassin...except that's exactly what the All Caste did. A week or so after leaving them he attacks a mob funeral. Then he takes out several mob families and takes their money before heading to Gotham. When would he have time for it if the LOA is crammed into this?! And how could he be so out of character to the point he was naïve about what the LOA did?!
*2 I know pre-Flashpoint there were other teachers like Shiva training Tim. I mean the long years worth of journey in preboot or in present canon.
*3 Jason returning would call a lot of attention to Bruce Wayne especially the fact that Dick Grayson was unmasked as Nightwing.