I'm not entirely sure what is required for streaming. I've heard that having a good internet connection with high download/upload speed is more important than your cpu.
Personally, I think that for overall multitasking the 965 will beat the i3, because on a true quad core you can assign your game to one or two cores and whatever you're streaming/recording with and your browser tabs to the other two cores.
The i3 has 2 true cores. True, each one may be a lot more efficient than any of amd's at the moment, but if you'll be doing some multitasking while gaming, I'd imagine the 965 pulls ahead, especially if you overclock it.
Nevertheless, it remains that if gaming is ALL you're doing (no fraps, no streaming, etc) the i3 remains unparalleled in performance. It should be stated though that the 3220 is still very capable of recording games (maybe not BF3 multiplayer, though). A true quad just does it better