It depends on the game, and monitor, but in just about all of them, the higher Instructions Per Cycle (IPC) on the Intel CPUs run circles around AMD. Most games won't take advantage of the FX-6300's extra cores, and the ones that do, will suffer more from the lower per-core speed.
That said, the AMD build saves enough money for a GPU that's double the speed. It will bottleneck a lot of newer games though. Again, it depends on the specific games here. MMOs and RTS's tend to take a bigger hit from an old CPU, while a FPS can be more GPU-dependent.
I'd probably go with Intel either way, just because it's a much newer platform, and it's easier to drop in a better video card later, than it is to ditch your mobo/cpu/ram.
I'd bump the Intel build up to a i5-6500 though. Higher clock speed for just a few bucks.
On the AMD Build, I'd swap a GTX 1060 6GB in for the RX 480 on the AMD build. Quieter, cooler, more efficient and slightly faster for about the same money. I'd look into that PSU too, not familiar with it.