There will be absolutely zero fps difference at 1080p. The only reason you need 4GB is if you're dealing with a HUGE resolution or an already-vram heavy game that's got dozens of mods. (skryrim with an 'ultimate' mod pack, for example.)
The only thing that the VRAM is used for is holding the image before it's sent to the screen. Just like with computer RAM, once you have enough, adding more gives you very very very little benefit.
That being said, are you sure about getting a 680? You can probably find a 670 for a lot less, and the 680 is only 5% more powerful than a 670. The 770 is more powerful than either of them.