Tautology: a = a
Theorem: a + b = b + a
Thinking of Godel, the tautology is the inherently unprovable statement that allows completeness, that is, it's an axiom. To Kant (if I'm reading the Prolegomena
right), a theorem can be presented in two ways: a + b = b + a
(analysis or restatement of subject in predicate) or a + b = c
(synthesis or the predicate being a wholly different concept derived from the subject). For example, the statement A line is the shortest possible distance connecting two points
is a tautology (Euclidean axiom) whereas the statement A line is a sequence of points extended in one direction
is a theorem based on said tautology (note lines can be either finite or infinite).