Sorry, the question makes no sense to me. A few points, in no particular order:
We may consider models of a classical first-order theory $T$ in any Boolean category. So there's really no sense in contrasting model theory with category theory; the former studies models of first-order theories in $\mathbf{Set}$, while the latter gives you the tools to study first-order models in much more general contexts. I don't see the contrast; as others have said, these are apples and oranges.
If we're going to study a theory $T$, then morphisms between models of $T$ should be assumed to preserve everything that can be expressed in the language of $T$. For example, if $T$ is an algebraic theory, we should consider homomorphisms between $T$-models. If $T$ is a first-order theory, then we should consider elementary embeddings between $T$-models. In fact (although I admit to not having much knowledge in this area), I think that category theory usually chooses the morphisms for you, automatically. For example, if $L$ is an Lawvere theory and $X,Y : L \rightarrow \mathbf{C}$ are models of $L$ in a finite product category $\mathbf{C}$, then a homomorphism $\varphi : X \rightarrow Y$ is just a natural transformation $X \Rightarrow Y$. We don't really get much choice in the matter!
Let me expand on the above point a little. If $T$ is a first-order theory in the language of $\{\in\}$ and $X$ and $Y$ are $T$-models, then there is nothing natural about functions $f : X \rightarrow Y$ satisfying $x \in y \rightarrow f(x) \in f(y)$. Remember, $T$ is best viewed as a first-order theory, not as some particular presentation for that theory! The fact that we decided to axiomatize $T$ using the signature $\{\in\},$ rather than $\{\subseteq,x \mapsto \{x\}\}$ or something else entirely, is irrelevant, and in any reasonable definition of "first-order theory", the theory $T$ would not "remember" how it was defined. So we really do have to preserve all the structure that $T$ can express; we don't get a choice in the matter.
Let $\mathsf{AlgGrp}$ denote the algebraic theory of groups. Let $\mathsf{1stGrp}$ denote the 1st-order theory freely generated by $\mathsf{AlgGrp}$, whatever that means. Write $\mathbf{AlgGrp}$ and $\mathbf{1stGrp}$ for the corresponding categories of models in $\mathbf{Set}$. Then there is a forgetful functor $\mathbf{1stGrp} \rightarrow \mathbf{AlgGrp}.$ It is faithful and surjective on objects. But it is not full.
On the other hand, if $T$ is an algebraic theory and $F(T)$ denotes the coherent theory freely generated by $T$, then (someone correct me if I'm wrong here) the corresponding categories of models in $\mathbf{Set}$ should be equivalent.