What makes an axiom an axiom rather than just an assumption?

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Are axioms discovered or invented? Why the axiomatic method took over geometry long before it reached other fields What really changes when you swap ZFC for another set theory Gödel showed any axiom system strong enough for arithmetic has undecidable statements The axiom of choice sounds obvious until you hear the Banach-Tarski consequence Does a mathematical system need axioms that are true or only consistent? Choosing between divide and conquer and working backwards What is the actual point of learning proofs when software can solve the problems? Gödel's incompleteness theorems explained without the jargon The noisy channel coding theorem in plain English