I would totally go with the FX-8320 over a i3-Anything. I just went through helping a friend upgrade his i3-3220 to a i5-3570 because the dual core wasn't fast enough for most of the games he was beta testing. It set him back about 350 bucks since he didn't feel comfortable installing a CPU and had a professional do it.
8 cores vs 2 cores there is no comparison. Sure a dual core will run your games but that's it. You won't find it able to multi-task much, like stream games on twitch or watch movies on a second monitor. If you go with a dual core i3, you will be upgrading it right away to a quad core once you figure out your system.
New games in the future will take advantage of more cores, I would consider a i3 to be baseline for today's games. BF4 isn't going to run awesome on a i3. It will take full advantage of a FX-8320 8 cores though. Also for your budget, AMD mainboards are generally cheaper and have more features than Intel mainboards. On the high end intel has better offerings, but at 700$ I would go with AMD.
I have a A10-6800k and use it in my lan party machine. I had a hard time even with water cooling to get the cpu temperatures under control. So what I did was I just turned off the turbo frequency in the computer bios and that fixed everything.
It runs stable at 4.1 ghz and never goes above 50 c.
I am running this:
CPU - A10-6800k @ 4.1ghz
MB - GA-F2A85XN-WIFI
8 GB of ram
R9 270