I am not completely sure I understand the distinction you are making between "axioms" and "assumptions", so this response may be missing the mark, but here is my interpretation of the question.
Consider the following example of a theorem in, say, ring theory:
Let $u$ be an invertible element in a ring $R$. Then... (something something something)
Now the assertion that $u$ is an invertible element is an assumption of the theorem; it is the hypothesis that logically entails a conclusion. Not all elements in a ring have multiplicative inverses, but for the sake of this theorem (whatever it is) we are thinking about an element that does have a multiplicative inverse.
On the other hand, every element of a ring has an additive inverse; that is built into the axioms of a ring, and as soon as we say we are thinking about a ring those axioms are implicitly invoked. So the fact that $-u$ exists is not an assumption, it is an axiom.
If this is the kind of distinction you have in mind, then I would say that the difference between axioms and assumptions has to do with whether the assumptions are local and temporary -- the conditions under which a certain conclusion holds -- or whether they are global and stable -- part of the conditions that define the sort of object we are thinking about.
There is of course some gray area in between. If we are studying general rings, then asserting that $R$ is a commutative ring might be a (local) assumption for a particular theorem or a particular set of theorems. But if we say in that specialized theory for a while, it might eventually become part of the background context and appear more like an axiom of the sub-theory.