A strong support character
You also get two new playable characters: Robin replaces Batman in Harley Quinn's Revenge DLC mission, and Catwoman is switched to every so often during the main game's story. She originally came included with new copies of the game or alternative required the purchase of a VIP pass -- a way for a publisher to make at least some money from the sells of used copies.
Catwoman felt so integral part of the story, that playing without her would definitely make the game a lesser experience. You can even get a faux ending at the end of her involvement if you try to leave Arkham City instead of going to help Batman. After beating the game, you can switch between her and Batman to pick up their respective collectibles you're still missing.
Freeflow combat fully realized
While Catwoman is a bit slower at traversing the City than Batman (due to not having a cloak to glide with), combat with her is more entertaining. She is quicker and her attacks feel snappier. The only real issue compared to Batman are enemies with fully automatic guns. More than one of them in an encounter and even on Normal difficulty you end up dying really fast thanks to Catwoman's low health. Stealthy approach is a must in those cases.
In my opinion they should have kept assault rifles in lower numbers in this game. Even as Batman with his higher armor and many gadgets, I often ended up using smoke bomb to get away to restart the fight. Combat gets too hectic to handle with guns in the mix of the already many types of foes the game has.
A few good puzzles
Arkham City's map is bigger than Asylum's. Instead of three open-air areas, there's one large area that connects all the locations and dungeons together. Riddler of course had to fill them all up with collectibles. The take-a-picture-of-some-meaningful-thing and look-a-question-mark-from-the-right-spot challenges I still didn't like but the new trophy puzzles I found to be high quality content. The only problem with them is that a player can try to solve one way for too long without realizing Batman doesn't have the necessary gadget yet.
This time Riddler didn't leave maps for his challenges. Instead you can interrogate mule thugs for the locations of the collectibles. If the mule happens to be in a pack, you have to dispatch everyone else first. I thought the mules were another good idea they implemented, although I'd say one interrogation should reveal more locations than they do as you end up doing it quite a few times.
I don't have much of an opinion about Arkham City's writing. I think it was largely the same as Asylum's. There is bit of an dissonance going on with Batman getting poisoned yet you not being in any actual hurry to get it cured since it will wait until the next story mission.
The ending was surprisingly heavy, though. Credits started rolling without a song (just some ambient) and then a voice mail from Joker of him singing Only You (And You Alone) kicked in. That was when I finally became convinced Mark Hamill is actually a pretty good voice actor.