I absolutely loved both games.
However, Deus Ex HR's boss battles were undoubtedly crummy, especially if you spec your character more toward stealth, like I did. But I'm a sucker for stealth games (Metal Gear Solid games are my all-time favorite games), and everything in the game that's not a boss battle is fantastic to play stealthy. I haven't played it otherwise, but I imagine it would be pretty decent to play run-n-gun or "tactical" as well. Its great stealth made it one of my all-time favorites.
I liked Arkham Asylum a lot, and I thought Arkham City would be a bit better, but wow. It was, somehow, way better (in my opinion, of course). It worked its way up to be one of my other all-time favorite games. Everything about it is just awesome, to me. I completed every task in the game, except for all of the Joker riddles (just way too many of those to keep being enjoyable, but they are fun and challenging to do occasionally). Arkham City also has its share of stealth, which also plays pretty well.
Overall, I vote for Arkham City as the first one to play, just because I like it more. However, they are both fantastic games, in my opinion, and you should play both!