Does the 2GB change how my games perform? since im using a 27'' screen....
And does the 560ti overclock better than the 6950?!?!?
I really wish that there was an obvious choice but they are so even...
The 560Ti overclocks well, but you can flash a 6950 to a 6970. 2 GB isnt needed for dedicated vram unless you are using a multiple monitor setup or a resolution higher than 1920x1080. I say go with the 560Ti, quieter, and cooler.
Yet flashing the bios has become increasingly more difficult because AMD has started to create read only bioses that cant be tweaked. You will find it difficult to find a card that will flash, unless you decide to go reference, which means you will get more noise and more heat than a non reference card.
Secondly, you don't want to flash the 6970 bios onto the card, because that will brick your card in the long run. You want to flash it with a custom 6950 bios that only unlocks the dormant shaders on the card (up to the same amount as the 6970). This prevents all the other stuff from going bad with a 6970 bios on a non 6970 card.
Thirdly, you are much better off just not worrying about flashing and just overclock the card to 6970 levels if you really need more performance out of it. Unlocking the shaders will also cripple the overclockability of the card, so you will get the same result in the end.
At that resolution (screen size doesn't matter) I would also suggest the GTX 560ti, however I would go with the MSI Hawk 560ti. Defenitely one of the best in that series and I have heard of stability issues with the gigabyte card, so I personally wouldn't go that direction.