I would use shown in this way, "He had shown them something" and I would use showed in this way, "I showed them that."
By the way, what is the "had" part of my first sentence? I can't remember it from my highschool french class (where I learned more about the english language than I ever had from my 13 years of English education in the American public school system :? ).
Also, in Exodus 33:18 it says, "And he said, I beseech thee, shew me thy glory." (I think this is Moses to the Lord)
Trick question, I say!