Let's suppose classical propositional logic (model theory) and Hilbert's axiomatic system (proof theory). I know that in the Hilbert's axiomatic system there are inference rules for deriving new theorems from axioms. By those inference rules it is also possible to check afterwards whether the given formula follows from the axioms or theorems. On the other hand, I know that in classical propositional logic there is a rigorous way of checking whether a particular formula $A$ is a logical consequence of a theory $T$ (meaning set of formulas). By a rigorous way I mean checking based on the defined logical functions (meaning logical connectives) whether a formula $A$ is true in every possible truth assigment of formulas of a theory $T$. Although I'm not sure whether there is in the classical propositional logic some "recipe" as well for deriving logical consequences from the theory $T$. Or is it that in the model theory there are only "rules" for checking whether a given formula follows from the theory? If so, is this the fundamental difference between a proof theory and model theory?
I should note that I'm quite a beginner in this subject, so easy on me please.