A 560 is faster IMO, plus it comes with 2gbs of ram. Note, though, that it doesn't have a full HDMI port. And some people give it bad reviews. You'll also be looking at a rather small oc IMO, as Fermi cards run hot, even with those fans.
You should consider bumping up to a $200 budget, for which you'd be able to buy a 7850 2GB. It's way faster and power efficient than those two. It also runs cooler, so a decent oc is easy.
Actually, i believe the 6870 is faster, in general (though not by much). It does have less memory, but i have that exact 6870, and i dont think ive encountered a framebuffer cap yet. Also, i can attest to the fact that the sapphire card is damn near silent and cold as ice under load. Some coil whine at ridiculous framerates, but its not a problem. Though id have to agree with your idea of getting a 7850. Ive heard those things are OCing powerhouses. 860/4800 stock clocks, going all the way up to about 1275/5800.