I think both words mean a sort of crevice or corner. Presumably there must be some significant difference.
The words are almost always used together:
e.g.
She searched in every nook and cranny
Presumably they can be used separately. I think I may have heard "nook" on its own, but never "cranny".
Interestingly the N-gram seems to suggest "nook" is more commonly used than "cranny".