Terminology is fluid (as Carl Mummert remarks). But in many/most people's hands, axiomatic vs non-axiomatic and informal vs formal mark orthogonal distinctions.
You can have informal and formal axiomatic systems (Euclid vs first order Peano Arithmetic). You can axiomatic vs non-axiomatic systems, whether fully formal or otherwise (e.g. axiomatic and natural deduction systems of first order logic).
Axiomatization is a matter of how some theoretical apparatus is organised: do we lay down some "starter" propositions, and then some rules for deriving more propositions? Or do we, e.g. regiment just using derivation rules?
Formalization is a matter of how stringent we are in specifying that apparatus -- usual informal mathematical standards of rigour or the sort of specifications we could (in principle) feed to a computer for formal checking?