The i5-4570 is a better CPU, but it also costs more. However, the i5-4570 costs almost $200 so at that point I'd say get the i5-4670K for $220.
If your budget was on the LOWER end ($600 to $700 or so) a case could be made for the FX CPU. I often recommend the FX-6300.
The reason for this is a good gaming PC (I'm assuming it's a gaming PC) is all about BALANCE. If you spend too much on the CPU then you run out of money for the GRAPHICS CARD.
*If you want better advice, give a BUDGET, say what the budget is for (case + internals + Windows) and listen to advice.
For people with a sufficient budget I often recommend the i5-4670K.