the 6950 performs better by a long way, while the 560 is cheaper and cooler.
The 6950 is better I would say but not by a long way. It beats the 560ti in most benchmarks by a few fps (around 7-13). In games optimized for Nvidea, the 560ti will beat the 6950 without a problem by about the same margin. Don't take into account the 2gb vs the 1gb differences in memory. the 2gb will only PARTLY be used when gaming in resolutions above 1080p, at which point the 6950 is a clear victor by a mile. Otherwise, the memory aspect of the two cards performs exactly the same.
Its really up to you which way you want to go. If you like Nvidea, go with them. The 56ti is generally cheaper by around $10-20, while the 6950 is about 5-9% better depending on the game you play.
As a side note: the 560ti is known to be a very very good overclocker, reaching speeds upwards of 1000mhz, while the 6950 has a (small) chance to be unlocked to the performance of a 6970($350 card). However that comes with its own risks as well.