I say get both in this case.

Highly debatable.  Both are open world,.  In my opinion, City has a larger world,and more villains while Arkham has a smaller world that's more detailed but offers better  gameplay.

What City does right is that it has more villains and some actual unique bosses, something Asylum doesn't have at all.  What it doesn't do right is that because of the grappling hook, they don't add any platforming portions into the game aside from the beginning when you play as Bruce Wayne.  You just use your grappling hook to get everywhere.  In areas that you're locked in with henchmen, where you can grapple to literally any railing so there's far less of a chance that you'll get seen by an enemy since you can grapple away literally anywhere.  Outside in the city itself there are a lot of reused buildings.  Literally constantly.  Constant travel back and forth from places means that you won't be able to get past certain points without grappling and as slow as it is to walk back and forth between those places will just make you grapple back and forth regardless.  Additionally, the Old Mill, The Museum, and Wondertower which consists also of the Subway and Underground are the largest areas in the game, while each area in Asylum is as big as any of those areas.

Arkham Asylum is open world but it's more contained, more detailed.  You don't have a huge city that constantly reused building.  Each and every area in the Asylum, whether it's outside or in is detailed.  The grappling hook doesn't work on anything.  It actually doesn't work on anything except gargoyles for most of the game.  So in elevator shafts, or certain areas that are vertically scaled, Batman has to platform and climb his way to the top.  In areas that henchmen are walking around trying to kill Batman, he can't just grapple to any ledge.  He can't even grapple to a gargoyle unless he's close enough.  This means that you're going to be down there with the henchmen actually having to use different ways to get around, or sneak around to behind the henchmen in able  to get them. 

Combat is mostly the same, with a couple extra enemies or weapons added on City, and couple of tools added, with the removal of the triple batarang and the triple batclaw.  I actually liked using the batclaw to pull down weak walls, another thing not included in City.

I love this series. Buy the first one and enjoy it, they play the second.

I liked Asylum a bit Better, it had more of a Metroidvania feel to it which I liked. And even though City was more open you still didn't explore Gotham. You just explored a bigger more open arbitrary island. I thought the story was a bit better in Asylum.

Play whichever one looks more interesting or both. The stories are spoiled a bit if you play City first, but I the plots in both games are quite predictable and doesn't get near as interesting as the animated series did. The character interactions with the VA's from the original series is the more interesting thing to me. But, the story is still quite enjoyable.

Thanks for the advice everyone, I just bought Asylum for $10 on xbox live

10$? Buy it.

You should get both games while they're on sale since they're both great.

IMO, Asylum had a better story, character development, focus, & atmosphere, while City had better gameplay and replay value. 

What did you think of the game so far? Both Arkham games are probably among my top ten favorite games from this gen.

I've only played for 20 or 30 minutes but I'm really liking it so far. The power that comes from picking your opponents appart with stealth is very gratifying and batmanesque. I'll play some more tonight and let you know how it goes.