the difference is the amount of vram the card have to save the textures it uses,
too little vram and you will get FPS drops and perhaps game freezes (just for a half a sec or so) when the graphic card is loading different textures from the harddrive.
currently if you have a 1080p screen you want more then 1GB, 1.5GB or 2GB is what you would like then
under 1080p for example 720p or such then 1GB is enough.
For example battlefield 3 in 1080p on ultra uses about 1.5GB vram.
The performance do not change, its still the same card however the difference is the vram (how many textures it can have loaded in to its memory at any given time)
so under 1080p resolution 1GB is fine, 1080p or higher then 1.5GB or 2GB would be helpful
ofc that depends on the card too, a card will not be faster becouse it have more vram, like a 6950 1GB vs a 6950 2GB will still performance the same in lets say 720p resolution, but if we go up to 1080p and a game that uses a lot of vram (over 1GB) then the 2GB card will run better.
most modern graphic cards have 1.5GB or 2GB vram now as well its needed these days as 1080p (and higher resolutions) become more and more common.