A mathematical proof is a step-by-step demonstration of the truth of a mathematical theorem. Proofs build on axioms, which are statements that are assumed to be true without proof, as well as previously-proved theorems.
Unlike the theorems of mathematics, science does not seek to prove that its theories are true. Instead, the scientific method seeks to check whether the predictions implied by a theory are observed in nature. Therefore, as philosopher of science Karl Popper argued, science can only hope to show that a theory is false. But scientists recognize that science can never prove that a theory is true in the same sense that a mathematical theorem is true. Therefore scientists never claim that their theories are facts. Instead, science searches for theories that are not disproved by currently-known experimental observations. Insofar as theories are consistent with nature, they may serve as a guide to improve technology for example and can be considered as true in laymen's terms.
In American courts, crimes are proved "beyond reasonable doubt" to a jury, based on the jury's own analysis of the admissible evidence. Other legal issues may be decided by clear and convincing evidence or by a preponderance of the evidence (more likely than not).