Although we call the rules which a space needs to be as a vector space as vector space Axioms. I find it that the Axioms are meant in a different way than the conventional sense of Axioms used in theories like ZFC or Euclidean geometry. The reason being that, in practice, we use the vector space axioms to check whether some specific space is a vector space or not while in ZFC or Euclidean Geometry the objects which are discussed are specifically defined in the Axioms itself.
How can this difference be described as a difference between formal systems?
Now, as far as I understand, you're right: While in both cases we speak of 'axioms', in the case of vector spaces this is just a way saying how a vector space should behave. The theory of vector spaces cannot be expressed in the 'axioms' of vector spaces, if there is not a logic involved also. You will need a 'language' to speak about all sorts of things..like set theory. And the 'axioms' of a vector space will not be necessary, as you can define what a vector space is, in set theory. In set theories instead 'axiom' seems to mean more something like foundational assumption, that is axioms and inference rules 'really are' (make up) your theory. As they can express (as far as I know) almost all of mathematics, unlike the 'axioms' of vector spaces, you can see them as a possible 'foundation' (argumentative assumption) for almost all of mathematics.
Hope this helps a bit, probably there will be more professional and in-depth answers soon.
I am not sure if I understand your final question. There is not really anything to compare as far as I understand, because, as said above, the 'axioms' of a vector space simply do not constitute a theory.