Ah, the age-old question.
Between the two processors you're looking at the FX-6300 would be my choice, particularly as the 3240 socket has been replaced, meaning that your upgrade path will be very limited. One of the newer Haswell i3 processors would make more sense, since the upgrade path is better. In that case, I'd suggest you go for the i3 or preferably jump straight to an i5.
I'll probably get a torrent of backlash for saying this, but Intel processors are more powerful than their AMD equivalents. They're also built on a more efficient manufacturing process so they consume less power and produce less heat. The upside to AMD is that all of their processors are unlocked, making them easy to overclock. You'll pay a premium to have this on an Intel processor.