Earlier quoted context omitted.
Proof of work relies on computations that are hard to do but easy to check that someone has done. The entire network has to check someone's proposed result. There are very few computations that have sufficient asymmetry to the degree needed to make bitcoin work, and the types of things that SETI@home needs don't fit that bill.
Isn't Folding@Home trying to run an asymmetrical computation, though? The broad strokes as I understand them are that you have a sequence of amines, you arrange them into a three-dimensional protein (the hard part) by whatever method, and then you evaluate the energy bound within your structure (the easier part?). Lower energy is better. The obvious problem is that there's no judgment of "correct" or "incorrect". Ana…
Proof-of-work works because a theoretical attacker would have to invest huge resources into an attack. For bitcoin it's trivial to say "for attack scenario X an attack has to spend at least Y", and as long as the expected reward is lower than Y the network is safe (that's for example one factor you use to determine how many confirmations you wait before accepting a payment). If the attacker could conceivably gain money outside the cryptocurrency for performing the proof-of-work, that makes the whole calculation less predictable, and likely exposes you to more situations where an attack is actually profitable.