Short answers, pulled from the story.
A theorem is a statement that has been proven through logical argument derived from axioms using specific inference rules within a deductive system. It is not merely an assertion but requires proof to earn its title as a proved result.
The National Council of Teachers of Mathematics published a book in 1940 that included at least 370 known proofs for the Pythagorean theorem. This publication illustrates how a single geometric fact becomes a theorem only when it has been proven through logical argument.
Scientific theories are falsifiable and make predictions about the natural world that must be testable by experiments, whereas mathematical theorems are abstract formal statements whose proofs cannot involve empirical evidence or physical experiments. Mathematics derives truth from logical deduction alone while science relies on data collection and observation.
The Poincaré conjecture remains generally referred to as a conjecture despite being proved in 2002. In contrast, Fermat's Last Theorem was historically called a theorem even though it remained only a conjecture for centuries until proven.
The four color theorem represents a prominent example where proofs rely on computational searches too long for human verification. Initially many mathematicians did not accept these forms of proof because they could not check correctness by hand alone yet computer-generated proofs have become more widely accepted.