Not only the best Batman game, but one of the best games, period.
Graphically, it's a masterpiece on par with Gears of War (they share the same unreal engine afterall). The lighting is dark and the mood is portrayed superbly. The game does a masterful job of immersing the gamer in the experience by paying attention to the small details. For example, as you progress through the game's many trials you'll start to notice Batman's suit actually reflects the strenuous battles he has been through (e.g., his cape will be torn, the suit will be cut, etc.). I even noticed he started to have a little bit of facial hair by the end of the game?! And the attention to detail doesn't just go to Batman himself, all of the characters are top notch. Though the henchmen are generic they still look great and distinct. The environments are equally impressive. A great feature in the game was the ability to scan certain objects which were related to (and would unlock) Batman lore. Very clever! It actually made the backgrounds more appealing and interesting as you try to solve the various riddler puzzles. And of course the music fits right in with the dark atmosphere that permeates Arkham Asylum.
What really puts Batman over the top is the gameplay. It has been so long since I have played a game with such a refreshing gameplay mechanic. Fighting is frantic, but fun even when you mess up. It's difficult to describe on paper but Batman's attacks are context sensitive and he will leap, block, and punch seamlessly as 20+ prisoners try to ambush him. It's a masterful gameplay mechanic that works brilliantly. In addition to hand-to-hand combat there's also some stealth action. You can drop from the ceiling to take out prisoners, strategically place and detonate bombs, do silent takedowns, and so forth. What's even better is the game's environment is open enough that every person can approach this in their own unique way.
Finally, the plot has a simple premise that is continually expanded upon throughout the game. Things start out well but continually go haywire throughout your adventure. Early in the game Batman seems to be almost complacent, thinking he will easily remedy the situation - but the adventure continually throws him curve balls up until the very end. While the plot is not particularly complex, it does set the stage for the game nicely.
I have played through the game on both the PS3 and Xbox 360 and didn't notice any differences (other than the added ability to play as the Joker on the PS3 version) so there's no reason for any gamer to go without this game, Batman fan or not.