Mercian :
The problem with the I3 is it is only dual core. Some of the newer games, I have heard, will not even run on a 2 core processor and this is likely to continue as newer games are released. I am not sure if the I3's hyper-threading would compensate or not.
This isn't accurate. Essentially all games will run on a dual-core, just recommended settings are usually quad-cores. In fact games can't actually tell the difference between hyper-threaded cores and real cores. Performance is lower on the hyper-threaded cores relative to a real quad-core CPU, but the game doesn't have any way to know the difference.
More to the point, even if such a limitation existed, the FX 6300 wouldn't help. These AMD CPUs aren't really 4, 6, and 8 core units. They are half that.
Inside of a traditional type of CPU are numerous pieces. Two of the main units are the floating point units (FPU) and the integer units. A single CPU core has one of each. AMD came to the conclusion that FPU was less important, and when designing these CPUs, they opted to mix this 1:1 ration up, by placing one FPU for every integer unit. So two of AMD's CPU cores are closer to being 1.5 CPU cores.
Well things get worse for AMD, as they went for a design that emphasizes high clock speeds over IPC, even though it isn't capable of exceptionally higher clock speeds. The problem here is that comparing one AMD core to one Intel core isn't even close. One Intel core performance about twice as fast as an AMD CPU core.
Overall, the AMD CPU tends to perform a little better than Intel Core i3 when you are heavily multi-tasking, as it can run more threads (think of one thread, as a single application) at the same time. The Intel CPU on the other hand with hyper-threading would be running essentially four threads at the same time, but each of those threads would be working much faster. (This is incredible simplified, but gets the general idea across) Whenever you are only doing a moderate level of multi-tasking or you are playing a game, you will find the Core i3 to provide superior performance though.
Long story short, if either of them has a performance advantage, it would probably be the Core i3, but that performance advantage isn't so great that the FX 6300 doesn't get advised from time to time because, because it does sell for a good price, and still does reasonable performance.
You might try checking on the prices you can find for them and low-end Core i5 CPUs now. The prices on these units might of dropped a little early in anticipation of Skylake.