I live in EU and my battle.net account is set in EU.
I'm going to be on during some days in NY, so I was thinking in buying the game there (US).
What happens when I go back to EU and I put the key in my account? Can I download the EU Client? Or my key is linked to the US Client?
Diablo 3 is cross-region. It's the same as Diablo 2 and StarCraft 1. You select the region in-game and can create characters on that region, and change at any time (characters aren't available cross-realm). It's not like SC2 where you need 4 versions.
So, whether you buy the US or EU version, it differs not.
As an addendum to XSlicer's post, there is one restriction you need to be concerned about when deciding which region to register your game in: The real-money auction house.
While you can play on any region you want (each with its own set of characters, its own friend list, and its own gold auction house), you can only use the real-money auction house of the region in which you registered your game.
