fuse, n.; fuze.
A fuse is a wick or other combustible cord for an old-fashioned explosive. A fuze is for more high-tech explosives: it's a mechanical or electronic device used for detonations.
In a different sense, fuse refers to a component that protects an electrical circuit by preventing it from melting. Thus, the two clichés derive from the different senses of the word: blow a fuse from the electrical-component sense, and have a short fuse from the wick sense.