More VRAM doesn't increase your baseline fps, only a faster card (like a GTX 560 Ti, GTX 570, GTX 660 etc) does that.
However... the highest-end games like BF3, Metro 2033 actually use more than 1GB of VRAM when running on ULTRA settings at 1920x1080. The same is true for Skyrim and Crysis 2 when using their optional high-res texture packs.
In those games on ultra at 1080p, if you only have a 1GB card you'll suffer big fps drops because you run out of VRAM. But when running on high settings, there's no dropoff because 1GB is enough for that detail level.
So having 2GB helps when running on ultra- but that's only provided your card is strong enough to handle ultra at 1080p to begin with! You also need more than 1GB of VRAM when running multiple HD monitors.
So having 2GB on lower-end cards like the GeForce GT 630 and Radeon HD 6570 is pointless, because those cards aren't fast enough to handle gaming at resolutions where the additional memory would make a difference.
The GTX 560 is just on the borderline- it's fast enough to play BF3 on ultra at 1080p with FXAA, but not quite enough for MSAA at 1920x1080. To play on ultra with MSAA and still have acceptable fps, you need a GTX 570/Radeon HD 7850 or higher. So having 2GB on a GTX 560 can be beneficial, but only in specific circumstances. It's great for Skyrim, Crysis 2 and Black Ops/MW3. But since the GTX 560 is only strong enough to play the top-end titles with FXAA or without AA, the benefits of a 2GB model are limited.
http://www.anandtech.com/show/5541/amd-radeon-hd-7750-radeon-hd-7770-ghz-edition-review/17
A better choice than the GTX 560 is the 2GB Radeon HD 7850 for $185. It's more powerful overall than the GTX 560 Ti and Radeon HD 6950.
http://www.tomshardware.com/reviews/radeon-hd-7870-review-benchmark,3148-21.html
http://www.newegg.com/Product/Product.aspx?Item=N82E16814161405