What settles a claim beyond dispute. Proof is a higher bar than evidence and is claimed far more often than met.
Old French prove/proeve, from Latin proba (a test, a trial), from probare (to test, to approve); proof is what a test yields when the thing tested survives being tested.
What settles a claim past dispute, and a higher bar than evidence because it must survive the attempt to break it. Proof claimed without having been tested is only the older, softer sense of the word: a trial not yet run.