Like 4745 said, the memory wouldn't matter. And like einterlan said, the HD 7870 is a bit faster.
The GTX 560 is not powerful enough to take advantage of 2GB of RAM. At most GTX 560 2GB card will give you 5-7 extra frame per second but the HD 6870 will still be faster So go with the HD 6870.
Here's a review between them in an unknown site http://www.kitguru.net/components/graphic-cards/zardon/sparkle-gtx560-2gb-oc-edition-review/