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."