To prove means to show that something is fact, while demonstrating can be explained as just presenting something. They seem to be synonyms, but that's how they've been used as far as I've seen

They each have a range of meanings. In some situations they can be synonyms. After all, in geometry, the closing words of a "proof" are "Q. E. D.," an abbreviation for a Latin phrase usually translated as "thus it has been demonstrated." "Demonstration" has a cluster of meanings around the idea of showing something to someone else. The key concept is "show." "Proof" has a cluster of meanings around the idea of "logical certainty." A proof is so logically perfect that it is extremely convincing. Once you've been shown a proof, you almost have to stop arguing. "The Tesla S P90D is more fun to drive than any other car ever built." "I don't believe it. You have to show me." "OK, I'll give you a demonstration. Here, get in. Now try the 'ludicrous acceleration' mode." "Wow! That is amazing. Yes, you've proved it to me."