It often feels like a proof by contradiction is like making your point through a witty joke and the contrapositive is sitting down and explaining why the joke is funny.

Since the two approaches are logically equivalent, and since it is usually easy for the average journal reader to convert between the two --- the distinction for me comes down to style. Most of the arguments I have seen by contradiction are usually shorter, more intuitive, and elegant. Hence, I would usually favor proof by contradiction (it is usually the first proof I come up with...) unless the steps in the contrapositive are particularly insightful.