You will see an improvement. Arguably, the FX CPUs performs as well as Core 2 Duo / Quad CPUs at the same clock speeds. Therefore, a FX-6200 @ 3.8GHz vs the older Core 2 Duo @ 2.2GHz should perform 73% better when taking into consideration 2 cores. When more than 2 cores are used the performance will be even greater. Note that this is just raw potential performance, not actual performance. For example if you were to overclock your Core 2 Duo by 10%, that does not mean all benchmark performance will improve by 10%. It may be more like between 4% - 7% depending on the program being benchmarked.
Also note that there are very few games that can use 4 cores, BF3 being one of those very few games.