How to cut the non-overlapping part of 2 shapes in Illustrator CS6

This doesn't answer your question specifically (no "cutting" involved), but another less-permanent way to do this is with a clipping mask. Duplicate the circle and make sure it is above the art you want to remove.

Then, select both the circle and the art, and do Object → Clipping Mask → Make.

This will just "hide" the portion that you don't want to appear without actually cutting the artwork.

Before:

After

If you view the Outlines, you can see the rest of North America is still there, just hidden.

^{Side note: the Hawaii on your map is HUGE. ;)}