In German, both “proof“ and “evidence“ translate to “Beweis“ which only has one single meaning – if something is “bewiesen“ then it is true and a fact. In English, I see the word “evidence“ used a lot for stuff that is obviously neither true nor fact. I’ve looked it up and don’t really see a difference between the meanings of “proof“ and “evidence“ – can anyone please ELI5?
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Evidence establishes credence or likelihood. The more evidence you have of a certain thing/event, the more likely you are to believe that thing/event is true. But evidence is not proof and credence is not certainty.
A proof is usually (mathematics etc) a logical argument that is irrefutable (if the argument is done properly). In a sense, if you can prove something mathematically, then evidence is unnecessary – a “truth” has been demonstrated.
Now in normal semantic use, the standard for “proof” is less absolute and the word might be used with a weaker meaning. In law, for example, to “prove guilt” in a criminal case is to establish something “beyond reasonable doubt” (US centric). In science, physicists might use a statistical five sigma standard as sufficient to prove a discovery (5 sigma means that a result has less than 1 in a million chance of occurring if that discovery is not true)
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