Let me offer an explanation of the difference between truth and provability from postulates which is (I think) slightly different from those already presented. (Although perhaps close in spirit to that of Gerald Egars's.Edgars's.)
First of all, if we are talking about results of the form "for all groups, ..." or "for all topological spaces, ... " then in this case truth and provability are essentially the same: a result is true if it can be deduced from the axioms. (There is the caveat that the notion of group or topological space involves the underlying notion of set, and so the choice of ambient set theory plays a role. This role is usually tacit, but for certain questions becomes overt and important; nevertheless, I will ignore it here, possibly at my peril.)
But other results, e.g in number theory, reason not from axioms but from the natural numbers. Of course, along the way, you may use results from group theory, field theory, topology, ..., which will be applicable provided that you apply them to structures that satisfy the axioms of the relevant theory. But in the end, everything rests on the properties of the natural numbers, which (by Godel) we know can't be captured by the Peano axioms (or any other finitary axiom scheme). How do we agree on what is true then? Well, experience shows that humans have a common conception of the natural numbers, from which they can reason in a consistent fashion; and so there is agreement on truth.
If you like, this is not so different from the model theoretic description of truth, except that I want to add that we are given certain models (e.g. the standard model of the natural numbers) on which we agree and which form the basis for much of our mathematics. (Again, certain types of reasoning, e.g. about arbitrary subsets of the natural numbers, can lead to set-theoretic complications, and hence (at least potential) disagreement, but let me also ignore that here.)
In summary: certain areas of mathematics (e.g. number theory) are not about deductions from systems of axioms, but rather about studying properties of certain fundamental mathematical objects. Axiomatic reasoning then plays a role, but is not the fundamental point.
