When referring to proving that a theorem is true, I have seen both words used interchangeably. Do they mean the same thing ?
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$\begingroup$In English, proof and demonstration have different connotations, as Théophile points out. A demonstration might be an application or “illustration” of a method. I believe most native English speakers would use proof.
But in some other languages, the word cognate to demonstration is what is used for proof. The letters QED, commonly appearing after a proof, are initials for the Latin phrase quod erat demonstrandum. In French, they say ce qu'il fallait démontrer. But in German, the word for “to prove” is beweisen rather than demonstrieren (and prüfen is to test!)
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