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*.  Then there is an everyday meaning, where it basically refers to a collection of hypotheses to explain something, or even as a synonym for a single hypothesis: *"the criminal investigators developed several theories about how the murder took place".*

A theorem is a truthful statement in logic or mathematics. However, 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.  2 + 2 = 4 is also a theorem, but usually isn't talked about as a theorem because it isn't something general or revealing.

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.