I'm not very familiar with the AMD CPU line-up, but I think the A10-6800K and FX6300 use the same cores (Bulldozer), so the only CPU advantage of one over the other is going to be in clock-speed. You may find that the FX6300 clocks better than the APU but I'm not sure if (or how) having the GPU integrated into the CPU die impacts upon overclockability.
From a system perspective, you'd have the opportunity to upgrade the CPU & GPU independently of one another, and you could also get a much more powerful & capable GPU by going down the discrete route.
It's very much a horses for courses thing. From your explanation of what you want to do with the system an APU sounds like the perfect fit. You require neither massive CPU or GPU power, and the APU approach should help you to keep system cost and complexity to a minimum.