The i5-3570K is better for gaming although you won't see too much of a difference between the two in terms of performance. I'd say the i5 is the better of the two because it runs cooler and more efficiently, so all you'll need is a budget air cooler (such as Cooler Master Hyper 212 Evo) if you want to overclock to 4.5 GHz. However, any higher then you may want to consider a better cooler (Noctua NH-D14 or Corsair H100).
With the FX-8350, however, you'll need a quality CPU heat-sink to keep temperatures low if you want to overclock, this will end up costing more than i5 + budget aftermarket heat-sink.
Depends on what your budget is. The i5 is better but if you have a strict budget you are better to go with Amd and spend more money on a gpu.
Also the fps difference between the two are minimal depends on what game you play and the resolution of your screen.
The intels better for budget, overclocking at cooler temperatures without more fans or watercooling, and is better for some progggrams. Also has integrated graphics which is a life saver if you run out of money for gpu. Amd is better for multi tasking and overclocking more but with better cooling equipment and more power. However I would go with intel and if you dont think it will multi task enough get an i7 with hyper threading
The 8350 is better at one thing, video rendering, and that's it, gaming, even general multi tasking the 3570 at an equivalent OC runs away from the 8350 - Oh, and the 3570K also as a norm, can OC higher