if I were building a pc on a budget I'd go AMD any way, most games depend on a good gpu anyway outside of very few intensive games where AMD can play pretty decent too.
if I could do it again,idk,I'd prob get the i7 since it'll age very gracefully in the future due to the 8 threads,while probably using a fraction of the heat/power of an AMD chip,but on a budget I'd get an AMD APU and play on only 720p and be merry, I'm honestly happy with xbox/ps3 level graphics.
like how cheap bottom of the barrel cards can play on 720p pretty good,soon that'll be the case for 1080p ,I personally don't ever see myself bothering with a resolution of more than 1080p. Intel built in graphics can already do 1080p 30fps from stuff I've read, and their hd4000 could play stuff at 720p, which is what I did for a month.
at the end, if you have the money to play,sure intel is better,but on a budget I'd go AMD APU