My understanding of axioms is that they are self evident truths that require no proof, which in my mind is similar to a dogmatic belief in the sense that dogma is a set of beliefs or doctrines that are established as undoubtedly in truth.
Axiom is a statement taken to hold within a particular theory. One can combine the axioms to prove things within that theory. One may add or remove axioms to the theory to get another theory:
Dogmas are axioms of cultural, religious, political theories. Again, one may add or remove dogmas to get a new theory, e.g.:
The difference is that it is perfectly ok to handle different sets of axioms in, say, mathematics and prove a theorem in Euclidean geometry one day and a theorem in Lobachevskian the next - just remembering when the fifth postulate does or doesn't hold, but it's not considered acceptable to hold several sets of dogmas at once. Life of Pi provides an illustration of the controversy of such a stance (the main character is Hindu, Muslim and Catholic simultaneously and his brothers-in-dogmas don't like to share him with the competition).
I'm sure early geometers were much more religious about their axiom-dogmas than the modern mathematicians, but I have no proof.
An axiom is something that is self-evidently true; it is so obvious that there is no controversy about it. In mathematics, you just have to accept some very basic notions in order to avoid circular reasoning. These can't be proven, but they can always (and often very easily) be observed.
Example from Euclid's Elements:
Or an example from propositional logic:
A dogma refers to (usually a religious) teaching that is considered undoubtedly and absolutely true. It is something you accept without any direct observation; dogmas are accepted by faith only.
I should add that some people would say that there is no difference between axioms and dogmas, because 'self-evident truths' are in some sense based on faith; that is that you accept on faith that anything that seems obvious and self-evident is true. An interesting read on this subject is Wittgenstein's On Certainty. I also want to stress that I don't mean to say that an axiom is "better" than a religious dogma (or vice versa for that matter).