pure performance wise 4870 will be a bit faster than 5770. if you really want DX11 i suggest you to look up for better cards. my advice is don't get any card that's weaker than amd 6800 series if you really care about dx11. if you play at 1080p you might need stronger card if you want to crank up the setting while having decent frame rates.
Coming from a man who spent 2 years with 2 4870's in his rig, and several months with 3, the 4870 generates a lot of heat. It's ability to clock down when idle is rather limited. The 5770 will produce less heat and use less power.
But, the 4870 is still a formidable card. 256-bit Memory Bus vs the 128 on the 5770 will make a difference, but, as others have said, the 4870 is only slightly better as far as performance goes.
5770 is only about 10% slower in most cases than the 4870. I would go with the 5770, runs cooler and uses less power.
OP already owned 4870 so going for 5770 definitely not a good move if what he/she really after was to play games in dx11 mode. yes it is true 5770 is only a bit slower than 4870 but you have to consider the performance hit when additional dx11 graphic enhancement being used.
for 1080p 5770 will not going to cut it in dx11 titles unless OP wants to use the card just for the sake of lower power consumption and heat. personally i suggest OP to go with 7850 if he/she has the money for it. something like Gtx560 Ti should be fine already but you might not be able to get 60fps