I finished Arkham City a little bit ago with only 53% of everything completed. While I enjoyed the game, I still found Arkham Asylum to be the better one.
The city in Arkham City just feels so cluttered. While on the ground, it doesn't feel as interconnected as it should and while there is a little, there's hardly anything of importance on the ground either. The interior locations of Arkham City don't feel nearly as expansive as Arkham Asylum.
Now one thing I did like about the game is how the combat is much better this time around. Instead of enemies attacking one at a time, two or even three enemies can attack at once, but you can also counter up to three, or even 20 depending if you're at a certain boss battle. Most gadgets in the game now have a quickfire option for use in combat, new combat skills, etc.
Also the villians. A ton of villians have been pushed into the game as well, as either sidequests or into the game itself. The actual bosses within the game itself were also pretty awesome, and unlike Arkham Asylum, you actually fought more than two boss fights. With Arkham Asylum, the only bosses you actually fought were Joker and Bane.
So overall, the game does things better that Arkham Asylum didn't, but some things aren't as good. It's a great game and while not being my personal favorite out of the two, it's not much worse.