The distinction as I see it:

Come, v. 1. To ejaculate. 2. To experience orgasm.

Cum, n. 1. The physical evidence left when one comes.

"Come" can be conjugated; "cum" is post-conjugal.  Sometimes "cum" gets conjugated all over the place, but such usage is rather sloppy.

Notice how the cum/come distinction can be subtle: "Bob has come on his hands", meaning that he has ejaculated upon his hands, vs. "Bob has cum on his hands", meaning that he has semen-stuff on his hands, without explicitly stating whom that cum came from!

Footprints says re come vs. cum: your last line is fantabulous. You should expand it to a Seuss-like poem: "upon my bum, said wild Tom Plum, is quite a large amount of cum. I may be dumb, or drunk on rum, but where did all this cum come from?"
Ichiro2k3 says re come vs. cum: The 2000 edition of the American Heritage Dictionary seems to agree with this writeup.