There pretty much the same although I would go towards the 4300 because it's newer and it's 32nm so less heat and less power (I don't think it uses less power though since Vishera is pretty much Bulldozer, normally smaller transistors means less power ).
Why don't you get something like an i3? What games do you plan on playing?
Take this as an opinion, but I guess it's ok.
Look, most games benefit of 2 to 4 cores
Also, most games will run better with a good per-core performance CPU
That said, the obvious choice is the 965BE, but you can only push that CPU to 4Ghz
Now, if you allow my suggestion, i would take the 4300 any time. Why? It performs very similar to an i3 3220 at stock, but you can overclock the FX4300 to 4.4Ghz with a simple aftermarket air cooler. It consumes less power than the BE, it performs similarly, overclocks way better, and manages to take hand to hand against the i3 which is one of the most recommended budget gaming CPUs from the Blue side.
Anyway, here at Tom's there is an article on the matter, if I find the link i'll post it.
I would recommend going with the newer version FX4300 and Melik is right it should run cooler. The 965 is a 125w model and runs at 3.4Ghz. The FX-4300 is a 95w and a runs at 3.8GHz with a turbo boost to 4.0Ghz. Of course both are Black Editions.