CPU: AMD FX-6300 3.5GHz 6-Core Processor ($97.88 @ OutletPC)
CPU Cooler: Cooler Master Hyper 212 EVO 82.9 CFM Sleeve Bearing CPU Cooler ($22.99 @ Micro Center)
Motherboard: Asus M5A97 LE R2.0 ATX AM3+ Motherboard ($62.89 @ OutletPC)
Total: $183.76
CPU: Intel Core i5-4440 3.1GHz Quad-Core Processor ($169.99 @ SuperBiiz)
Motherboard: ASRock H97 Anniversary ATX LGA1150 Motherboard ($67.98 @ Newegg)
Total: $237.97
Both at stock settings according to Passmark multithread it's about even, the i5 at 6456, and the 6300 at 6444. The i5 is only stronger here by well... nothing. However, single core performance is where the i5 typically shines, which unfortunately happens to be a vast majority of titles out there, but that's changing slowly now, but a fairly long way to go still. In the single core Passmark score it's 1880 for the i5, and the 6300 at 1446. The i5 is 30% faster here, which even after an 6300 overclock, won't (most of the time) come close to the i5 in terms of performance.
I wouldn't say Passmark is the most accurate, however I've done tests using an i3 and 6300 that I own, and it was only off by about 1% during my testing in Premiere Pro CC. I didn't do any gaming benchmarks, because unless you play on a high refresh rate monitor, you won't really need a really strong CPU, in my opinion.
Overclocking the 6300, you are probably 10-15% off in terms of performance (better multi performance at this point), however, the price difference is pretty huge right now in my opinion, even including an aftermarket cooler for the 6300. The i5 build is 29% more expensive for probably a 15% performance increase after a decent overclock, which by the way the motherboard I picked I have personally overclocked on, it's fine. But it won't handle any high voltages very well, so the gap is even closer. At this point the i5 is simply a better choice, mainly because you won't have to overclock it, or worry about throttling/temperatures.
All the best!