Basically at higher resolutions more textures are loaded and in order to keep all these textures/high settings/AA to run smoothly you need more VRAM. If you don't have enough VRAM you have to turns things like AA down in order for the game to run smoothly. Like with RAM, more you have, more smoothly multiple programs will run at the same time. With more vram, more textures and detailed can be loaded and run smoothly.
A lot of games use well over 1GB of this buffer for various reasons (large textures, AA, backbuffering, high resolutions) but the point is, just because they're using more doesn't mean they are rendering more frames per second.
Basically, in a game, if a 2GB card uses 1.2GB and a 1GB card uses 1GB, the 1GB card probably won't perform any differently because it still has space to prepare and send out frames at the same rate.
in my opinion 2 gb one will be better.:)