For gaming purposes they are pretty close together. If I owned one or the other and only wanted to game I would not "upgrade" to the opposite. If I were to upgrade from either for gaming purposes I would go straight to Intel. Sometimes the 6300 does better(WoW Mists of Pandaria or whatever it is called seems to like the Vishera line), but many times they are very similar(Starcraft II for example).
Generally speaking the 965 is 60% of the cost of a 6300 but in general the 6300 is better so if you are buying new then the 6300 is best, if you own one or the other already then you're pretty safe staying with what you have. For gaming only you could probably go either way and not notice the difference unless you use a program to gather the FPS stats for you.
You won't get this significant overclock with a 620 either, especially not on the 6300.