Hi, thank you!
The antagonistic relationship that develops between Damian and Tim is largely due to both of them being extremely insecure around the time that they meet— with just cause. Tim was officially adopted by Bruce, Damian was brought to Bruce by Talia, and shortly after these changes in their young lives, Bruce dies (as far as the children all know). They both lose their father quite suddenly— Tim, soon after he’s cemented his relationship with Bruce (which is no easy feat because Bruce), and Damian, before he really gets a chance to know him.
Everything is in upheaval with Batman suddenly gone, and the position of Robin becomes more tenuous and uncertain than usual. Being Robin now means much more than a mask and patrol— it means a secured position in the family that just lost its leader, a family in which Damian has never known his place and Tim was beginning to feel stable in finding his own.
When Dick decides to fill the void left by Bruce and Batman both, Tim is appalled that this entails the Robin mantle being taken from him and given to Damian instead. Dick is Tim’s friend, his older brother, and it initially comes across as a betrayal. And for all his bravado, Damian must know somewhere in his stabby brain that he hasn’t yet done anything to earn this mantle (and no, being the fruit of Bruce Wayne’s loins doesn’t really count). Dick tries to explain it, but both boys are confused and strained by the recent loss and changes in their lives, and they see this passing of the torch as arbitrary at first. They know only that Dick now has the power to enact such a shift, and that he is now the person whom they must appease.
So they hate each other, and they fight: Tim, to prove that Damian isn’t worthy to take his place, and Damian, to prove just the opposite. It has little to do with each other personally, and everything to do with being terrified that they can each lose everything at any second. It’s easier to attach those fears to a specific person: to a face that can be punched when you need an outlet for your anger. So these two jump each others’ asses all over Wayne Manor and even on patrol once (cutting Tim’s line— Damian, are you serious), and for all that Dick tells them he isn’t impressed by it, Tim and Damian each seem to think that eradicating the other will solve all of their respective problems.
On some rational level, I’m sure they both know that isn’t true, and they eventually do get past this phase of constantly acting out. Then Bruce returns, and we go back to square one because now there’s a different Batman to impress. Bruce is not at all permissive with this kind of behavior, though, and tells them directly to cut it the fuck out (and with Bruce, unlike with Dick, there’s always an implied or else). So last I read (and I’ve been pissed and avoiding new issues since Damian was killed), Tim and Damian had settled for insulting each other with occasional snarky comments from afar. An improvement over a physical brawl upon each encounter, I guess.
Time doesn’t pass quite the same way on a comic book page, but I would guess they move past this grudge and begin to even understand the roots of it themselves as they become more secure in their current roles. I mean, Damian’s current role is dead, but DC is up to its usual nonsense and that may change soon. So in my head where I pretend Grant Morrison didn’t kill Damian for no good reason and DC didn’t allow him to do so, Red Robin and Robin slowly develop a begrudging respect for each other as Tim heads into his twenties and Damian becomes an older teenager. They settle into their roles and realize they have no reason to feel threatened by the other, and their squabbling becomes more of a brotherly sort. Maybe Tim even helps Damian with forging his own hero identity when Damian reaches the point that it’s time to pass Robin on to someone else.
Batman: Gotham Adventures #14
fhfadsjdfsjdf! Another epic Wunder Boy pose!
The Evolution Of The Batman Logo till 2012. Check out all about the Batman’s 75 Anniversary on the corresponding DC Comics page.