in terms of gaming performance,
radeon hd 6770 1 gb gddr5 > hd 6670 1 gb gddr5 > hd 6570 1 gb gddr5.
if the card has ddr3, it will perform lower than it's gddr5 counterpart.
2 gb vram is wasted on gfx cards with these gpus, they only add to price and add nothing to performance.
the 6770 (reference design) requires a pcie power connector.
6670 and 6570 (reference design) don't require any pcie power connector.
hd 6770 will win the hd 6670 and doesnt requires any pcie connector and it need a psu around 400watts but both will not give much performance as the hd 6770 can give look at the camparison you will see that the hd 6770 is twice of hd 6670.the hd 6750 is also better then the hd 6670.