Axioms are self evident truths that require no proof, which 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.
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:
Things that are equal to the same thing are also equal to one another (Transitive property of equality).
If equals are added to equals, then the wholes are equal.
If equals are subtracted from equals, then the remainders are equal.
Things that coincide with one another equal one another (Reflexive Property).
The whole is greater than the part.
Dogmas are axioms of cultural, religious, political belief systems.
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.
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. Take the axiom of choice, for example: there is huge division in mathematics as to whether it’s true or not, and many proofs are written based on a by-faith acceptance (or rejection) of said axiom.
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.