In a slogan, we might say (without being too misleading) that computability theory is about the finite, model theory mostly about the infinite -- a rather big difference!
In a bit more detail, computability theory is about what can and can't be done by algorithms -- at least in the basic, original sense, these are finite sets of instructions for operating on finitely specifiable data (natural numbers, or stuff that can be coded up using natural numbers, like finite strings of symbols). So computability theory starts by showing the equivalence of various sharpenings of that intuitive idea (e.g. the Turing and the Gödel-Herbrand and the Kleene characterisations of computability come to the same). And proceeds to prove limitative results about what can't be computed. And then proceeds to fancier stuff about computational complexity, etc etc.
Model theory is about the relationship between theories (bunches of sentences) and their models in which they are true -- typically, in the interesting cases (at least at the outset), these are infinite structures. And we are interested in results like the Gödel completeness theorem (syntactically consistent sets of first-order sentences always have a model), the upward and downward Löwenheim-Skolem theorem (if you have an uncountably infinite model, you'll have bigger infinite models, and smaller countably infinite models). And so it goes. Very different issues.