It may help clarify the issue with "axioms" by looking at how the meaning of that word has changed. In Euclid's Elements, and for a long time after that, an "axiom" or "postulate" was not just any sentence: an axiom had to be obviously true and self-evident, so that no proof was required. In this traditional sense, the negation of the parallel postulate would not qualify as an "axiom", because it's not obviously true. For example, the fuss over the parallel postulate started because it wasn't clear that the parallel postulate was sufficiently self-evident.
In modern logic, we worry much less about the "self-evident" requirement . When we are working in complete generality, any set $S$ of sentences can be regarded as a set of axioms. The set of all sentences that can be deduced from $S$ is then the deductive closure of $S$. With this reductive meaning of "axiom", there is no longer much difference between a theory and a set of axioms. We could consider every sentence in the theory to be an axiom, for example, while Euclid would not accept every statement provable form his postulates as a postulate.
The word "theory", as matt says in his answer, is used in several ways. Sometimes it is used to mean a deductively closed set of sentences, and sometimes it is used to mean just any set of sentences, which might also be called a set of axioms. In most settings, we can replace a set of axioms with its (unique) deductive closure when necessary, so the difference between the two conventions is not very substantial.
There is one more meaning of "theory" I want to point out. I mentioned theories that are generated by taking the deductive closure of a set of sentences that are treated as axioms. Another way to form a theory is to take some class of semantic structures, in the same signature, and then form the set of all sentences that are true in all structures of the class. For example, one can form the "theory of abelian groups" and the "theory of the real line". Such theories will always be deductively closed. The difference here is that we did not start with a set of axioms.
 We do worry about self-evidence when we are trying to justify a foundational theory such as ZFC. And the axioms we assume are often obviously true, in which case there is no issue. But we also look at axioms like the axiom of determinacy, which is disprovable in ZFC. Some of the traditional usage remains, but only in certain contexts.