Yes there will be a Difference between a 1Gb & 2Gb GPU....
For a single 1080p monitor, you do not need more than 1GB of memory on your video card. There are only a couple of games that could potentially use more than 1GB at 1080p (Crysis 2, Metro 2033, Grand Theft Auto IV), but those games aside from possibly Grand Theft Auto IV will be bottlenecked by the speed of any given GPU before the amount of RAM becomes an issue. These games would only run over 1GB when run on maximum settings, with very high Anti-Aliasing. In the case of Crysis and Metro, there is no single GPU that can do that. GTA IV is just so poorly coded that it requires something like 1.4GB of video RAM in order to run maxed out.
u will need a 2GB if you plan on gaming across multiple monitors, or you intend to keep your card for a very long time, in which case 2GB might be a little more "futureproof", though by the time games routinely start using more than 1GB at 1080p, most current cards would probably be obsolete.