As Jonny indicated it depends mostly on what resolutions you want to play at and and what games you play.
The cards are a very similar value and similar technology. The 5770 costs $30 more and is proportionally more powerful. So if you have the money get the better 5770.
EDIT: but make sure your power supply is strong enough.
@ Happyjesus: 5770.
My i5 750 is running one at 1920x1200 with few problems but I seldom use more than 4x AA at such a high resolution, if you want the full eye candy the 5850 is a better, if rather more expensive, choice.
BTW if your MB supports CF at 8x/8x dual 5770s' are a devastating combination, if not for now, then for the future .