Either processor would be able to fully utilize the 570. As far as required power:http://www.guru3d.com/article/geforce-gtx-570-review/8
You could get away with a 600W (I'd say 500 if the video card weren't OC'd) if you get the right one. This Corsair 600W is totally sufficient for either an i3 or i5 build with an OC'd 570:http://www.newegg.com/Product/Product.aspx?Item=N82E16817139019&cm_re=corsair_psu-_-17-139-019-_-Product
Otherwise, if you feel safer going with what Gigabyte recommends, a Corsair 650W at $20 more:http://www.newegg.com/Product/Product.aspx?Item=N82E16817139020&cm_re=corsair_psu-_-17-139-020-_-Product
A quality 600W supply as I've recommended will truly be sufficient, though. Gigabyte puts a higher wattage level as a recommendation in case someone's skimming the unboxed PSU bargain bin for the cheapest supply possible. By saying you need 650W, they figure they are covering most of the supplies out there. If you look at the power requirements, though, you'll find you don't need that much if you pick up a good-quality supply with sufficient amps and wattage on the 12v rail.