The 4870 and 5770 are very similar performance wise (over all). The ATI 5770 is the newer tech, and so will likely run cooler/more efficiently, and offers DX11 support. That would be the choice in my book.
Considering they can both be had in the $150 to $160 range, I'd say just go with the ATI 5770. Just make sure you get the standard 1GB version (there are a couple 512MB versions available).
XFX ATI 5770 1GB (Double Manufacturer Warranty) & Free Shippinghttp://www.newegg.com/Product/Product.aspx?Item=N82E16814150462&cm_re=ati_5770-_-14-150-462-_-Product