VRAM's major use is for texture storage and quick access to certain graphical aspects. For instance, instead of your computer running two miles to the nearest grocery store (HDD or normal RAM) for some chips (textures) it runs to the nearest gas station which is 1/4 of the distance (Your VRAM). The more VRAM you have, the more of these graphical aspects that can be stored.
Now in a realistic setting, the only thing you'd gain is a little performance like Nickelob said. In the long run though, you'd probably be better off with the 1GB version as it is more than likely cheaper (Unless you can afford the 2GB, then by all means go ahead.) The biggest thing about GPU's is their clock speeds. Might I ask some specifics on the card you're looking at?