A theory (in science) is a set of hypotheses which give a model about how something works. For instance, *Einstein's Theory of General Relativity*. Another meaning of *theory* in academics is that it is some area of study or knowledge, or a special branch: *music theory*, *graph theory*, *category theory*.

A theorem is a truthful statement in logic or mathematics, which carries some significance.

Truths which are taken for granted as the basis of a system, are also not theorems: they are *axioms*. An example of a theorem is the claim that the square root of two is irrational. This isn't obvious and requires a proof.

Theorems are proven absolutely by deduction.

Theories (of the first kind, scientific) are not true or false; they are judged by experiment: the degree to which the theory agrees with reality. Areas of disagreement do not invalidate the theory, but only establish its limitations. 

The slightest flaw in a statement presented as a theorem, however, such as a counterexample to its general claim, shows it to be a false statement: a non-theorem.