From a proof theory perspective, there is no difference. They are both effectively announce the truthhood of something without providing a proof thereof.
The difference arises when one applies model theory, which is required to apply the mathematical results (whether applied explicitly or implicitly). A definition is wholly contained within the mathematical system. One cannot disagree with it because it is simply an artifact of the way the system is written. One can also sometimes rewrite the system to exclude a definition which is "offensive."
An axiom, on the other hand, reaches outside towards the system that is being modeled. These axioms define the range of problems for which the mathematical systems are applicable. If one disagrees with an axiom, it simply states that the mathematical system is not applicable to a particular class of problems because you are not willing to accept the axioms.
From a practical perspective, there is some difference between writing a definition from writing an axiom. You have a little more freedom when naming and defining definitions, because you wholly control their meaning. When it comes to axioms, you tend to have to interact with what others define things to mean. As an example, within a mathematical system, I may elect to redefine "+" to have a meaning not usually associated with addition. This may be effective for visually depicting a concept and making sure the reader remembers it (so long as it is close enough to addition to not give the cognitive dissonance). However, if I provide an axiom which requires something be "continuous," and my use of "continuous" is actually not the same as the more agreed upon definition, now I can cause great confusion. The axioms are something which are typically addressed up front, before your own style has leaked into the notation and verbiage. If one uses a standard terminology in the axioms, it is more likely to confuse someone who is scanning across a bunch of papers looking for a solution to their problem.
A great example of an axiom shows up in physics: "a closed system." A closed system is one where no energy crosses the border of the system (derivative of energy flux is zero). This could be a definition in some abstract scenarios, but in almost all cases it is an axiom. Not all systems satisfy the "closed system" axiom (in fact, technically speaking, no system 100% satisfies it except perhaps the universe as a whole). The applicability of any mathematical modeling under the axiomatic assumption of a closed system is limited by how well "closed system" describes the system someone is exploring.
On the other hand, there could be cases where one would elect to use it in the sense of a definition. For example, if you were working with an abstract mathematical construct and you found a subset of this construct which has behaviors similar to a closed system in thermodynamics, you may elect to define a closed system to match that subset of your construct. One might be exploring a class of ring generators, and notice that some of them demonstrate a behavior like entropic decay. One may choose to identify these behaviors with thermodynamics terms like "closed system" because it does a good job of capturing the relationships you are focused on. However, since it is purely encapsulated within your mathematics, it's okay if it's not "the official definition." That definition does not have to interact with the thousands of papers on thermodynamically closed systems quite as much as you would if your construct was only applicable to closed systems. In that case, you would want to treat it as an axiom.
In all, it's effective to think of a "definition" as something internal to your work, while an "axiom" tends to connect to the greater body of work, defining which classes of problems allow the application of your work.