Arkham Asylum was okay, but it is pretty mediocre in retrospect...especially when compared to the successor.
Combat is pretty bland in the Asylum. Batman simply doesn't have enough moves in the first outing to allow for a whole lot of choice in how to approach combat. Normal mooks are dispatched through simple Punch+ Counter/Dodge game and the specialized foes mostly demand only that a certain "opener" be used on them. And some of the weapons enemies can pick up and use make certain fights feel like a chore...you've got to follow whoever might pick up a gun. The batclaw and batarangs can be used in combat, but neither are terribly handy nor interesting. The only real "special" move Bats has is the Instant Takedown which is useful, but a little bland; every time you hit a combo amount that allows for the use of a special move it's a no-brainer to input an Instant Takedown.
Arkham City was
much better in this regard. The Bat's moveset is practically doubled, with some important additions like more combat gadgetry, a Fist of the North Star beatdown move, and three useful special moves to complement Instant Takedown (and make deciding what special move to use after hitting certain combo milestones and actual decision). Disarm and Destroy is probably my favorite addition, allowing you to control combat scenarios without having to play gun-tag, but there's plenty of other things Rocksteady added to actually make combat fun...for popamole stuffies.
Asylum and City have pretty similar stealth sections and none of them are particularly challenging because the AI simply can't compete with the abilities of the Caped Crusader. Asylum is probably worse because of how easy it is to win practically every stealth section by doing nothing but camping gargoyles (vantage points), going for a quick takedown, and then grappling back up to the safety of the gargoyle. The devs try to stop this strategy bluntly in Asylum by rigging gargoyles with proximity explosives in a couple sections towards the end, but it still is pretty easy to switch over to vent or grating camping. City did a much better job of adding difficulty to stealth sections with Detective Vision jammers and foes that would actually destroy all the vantage points if you gave them reason to suspect you were up there. Still, the stealth sections (in both titles) are pretty easy sans post-campaign, challenge maps that mandate a ballsier playstyle to get higher scores.
Both games rely on collect-a-thons for the majority of extra content within the campaign. Asylum's collecting rewards the player for always having Detective Vision engaged and for returning to previous areas with late-game equipment to grab the goodies that are behind [GADGET X] barriers. Arkham City indulges in this design trope as well, but not to the same degree. At least half of the collectible trophies have micro-puzzles attached to them; things that actually ask for a little bit of thought in how you make use of the wonders within the utility belt. Makes collecting actually fun at times. Arkham City also has actual sidequests to do, none of them exceedingly good, but these help to throw in some second-tier villains as well as set up hooks for the next game's storyline.
And since both games are mainstream AAA AWESUMFESTS, they're heavily reliant on variety and set pieces. Arkham City is much better in this respect. Locations are far more diverse, both aesthetically and mechanically, in the City than the nuthouse and AC actually delivers more than one boss fight that isn't oh so.....Banal
.
TL;DFR version:
Arkham Asylum is bland popamole, Arkham City is "good for what it is" popamole. City is unquestionably better from a gameplay perspective, but it might not be good enough if you don't want to put up with the core popamole that runs through both games.
