By way of perspective, I have two systems that can run anything/everything I have. One has an i5-3570K, and the other has a Phenom II 970BE. The FX-6300 you are considering offers performance somewhere between them, probably closer to the latter. In any case, both CPUs run all my stuff. Games are more dependent on the video card, so depending on how those are switched around, my games run well on either system. That said, the i5 runs them a little faster, and a little smoother. It isn't extreme, but I can tell the difference. You won't suffer with either one, but consistent with benchmark charts, the i5 is undeniably faster, and it uses less power for the speed it offers.
Some notes of caution though, especially if you're on a budget. There are a lot of cheap AM3+ motherboards out there, with a wide variety of problems. You'll find a lot of older 700-series boards with 900-series names. They typically lack modern interfaces, such as SATA 6Gb/s and USB3.0. You'll even find some old nVidia chipsets, that in addition, run their hypertransport buss as slowly as 1000MT/s (2600MT/s is common on a modern board). These, and some of the cheap true 970-series boards (*cough* MSI *cough*) also have weak VRMs, that cannot overclock, and will likely pop if you try. AMD's FX chips really need to be overclocked to get anywhere near their performance potential, and that means you can't go too cheap on the motherboard, and you'll need an aftermarket cooler (another $30+). On a really tight budget, IMO, you'd do better getting a relatively inexpensive H77 board and an i3; you can always upgrade it to an i5 later.