Difference between consistency and satisfiability

Consistency is a syntactic property. It means that there is no proof of contradiction from your axioms.

Satisfiability is a semantic property. It means that there is a model of the axioms.

In first-order logic (as well as propositional logic) the two notions are equivalent because the logic is sound and complete. Meaning a satisfiable theory is consistent, and a consistent theory is satisfiable.

Other logics, however, are not so lucky to have both of these properties and the two notions separate. In fact, if we do not assume the axiom of choice, then it is consistent that there is a theory which is consistent but not satisfiable.