Proof and evidence sounds similar in meaning, Both refer to a court process, where a fact needs to be or has been accepted as truth for the purposes of the judgement in a dispute.
Evidence is a means of proving a fact. It may be a paper, an object or even a statement from a witness presented in the form of a testimony.
Proof, on the other hand, is something that has already been accepted by the court or public authorities as truth, typically be presenting evidence.