For video editing, I would go with Intel. That being said, you're going to pay a bit more for your setup. Not only are the CPU's themselves more expensive, but motherboards are too.
If you want flawless video editing, I wouldn't do anything short of a Core i7. ( For gaming this obviously isn't needed, any Quad Core would do ) The i7 would give you the horsepower you need to run such applications. Ideally, you'd want to do a workstation build, with an Xeon, but we're trying to keep costs low, so that seems to be out of the picture.
IMHO - up your budget $200-300 more dollars, and get something similar to this:
4-8GB of RAM ( DDR3, 1333-1600 with tight timings e.g 7-7-7-24. or 8-8-8-24 )
A single Nvidia GeForce 660ti would be more than enough to power any game out on the market, with reasonable settings. ( My 560ti still doesn't choke ) Anything higher would be out of budget, and an overkill.