It's hard to choose for me because I feel like the things they do well are completely different from each other.
The Nolan universe is great cause what it really showcases is the psychological/emotional underpinnings of Bruce Wayne, and what would cause him to want to become Batman. The Dark Knight trilogy isn't realistic in the sense that the events that happen in them are all plausible in real life, cause they aren't, but it's "realistic" in the sense that the characters' motivations and development are grounded in the real world. It's why, for all the complaints it may get for being "out of character", it makes perfect sense for Bruce Wayne to retire from being Batman at the end. No person would ever be able to keep going on as Batman for long. And I think that drives home the "realism" that Nolan was going for.
It's not about making sure every single scenario that plays out is perfectly explained by real-world logic, it's about taking the Batman franchise and its characters and placing them in a grounded world and having them act out their story under a veil of realistic human characterization.
The Rocksteady games, on the other hand, are amazing for how fleshed out the world of Gotham is and how much Batman fan service they manage to cram together (even if it gets too feeling convoluted at times). Like others have said, it's basically a love letter to the Batman franchise.
As an aside, I have actually never read a comic book in my life, they just have never interested me, so I'm not too familiar with what's "in character" or "out of character". I'm aware of some of the more famous story arcs and such but never looked at them in much detail. The Nolanverse and Rocksteady games are pretty much my only two exposures to Batman. I appreciate the Nolanverse for making me fall in love with the character in the first place, and I appreciate the games for familiarizing me with the huge amount of background and lore of the franchise, and for translating the characters' depictions straight from the comic books, so I can see how they "really are". (And I admittedly have less opportunity to become bothered by things in both universes, due to my lack of comic book experience).
So maybe I'm not even fit to answer the question of "who did Batman better?". So I won't answer it. But I believe both are fantastic portrayals of the character. And they have allowed me to deeply enjoy a comic book franchise that I wouldn't be exposed to otherwise.
