Category of groups = Category of models of group theory? Is the category of groups with group-homomorphisms the same as the category of models of group theory with elementary maps? 
If not so: why?
 A: The Categories will be fundamentally different. The category of groups with group homomorphisms, (even with monomorphisms) enjoys a directedness property: any two groups can map monomorphically into their direct sum. 
But in the category of models of group theory under elementary maps, the finite groups map elementarily only into isomorphic copies of themselves. The size of any particular finite group (or indeed any model in any theory) is first order expressible, and any elementary map will be an isomorphism.
A: Let me repeat my above comment as an answer, since I think it is important to the discussion.  (I agree that JDH's answer is the answer to the specific question asked, and what I write below assumes that you have read his answer.)
If you remove the word "elementary" from the question, then it is indeed true that the two categories are "the same", in the strongest possible sense: they are canonically isomorphic.  (The notion of equality of abstract categories is well known to be sticky and unfruitful, just as for the notion of equality of abstract objects: c.f. Mazur's wonderful essay "When is one thing equal to another?")
Given a language L, one defines L-structures [or slightly more precisely, relational structures] and morphisms between them [often required to be injections, but let's not do so here]. This is certainly a [concrete] category, even though for some reason it is not standard to say so explicitly in Chapter 1 of model theory books. If you have a theory T of that language, then it is natural to consider the full subcategory of models of T. If you do this with the theory of groups [in the language of monoids], what you get is a category in which the objects are groups and the morphisms are homomorphisms of groups.  In other words, you get back [up to the provisos of the previous paragraph] the category of groups!
A: I'm not sure if my answer is to the question you're asking (perhaps you can rephrase/extend it? e.g. by providing definitions that you do know, so it's clear which ones your asking about?).
Consider the category $\mathbb G$ with categorical products freely generated by an object called $G$, a distinguished morphism $e: 1 \to G$ (where $1$ is the terminal object) and a distinguished morphism $m: G\times G \to G$, modulo three relations:


*

*The two maps $m \circ (\text{id} \times m)$ and $m\circ (m\times \text{id}) : G^{\times 3} \to G$ agree.

*The three maps $\text{id}$, $m \circ (\text{id} \times e)$ and $m\circ (e\times \text{id}) : G \to G$ agree (where I have identified $1 \times G = G = G\times 1$; really I should include the canonical isomorphisms).

*The map $(p_1 \times m) : G^{\times 2} \to G^{\times 2}$ is an isomorphism, where $p_1$ is "project onto the first factor".


Then the category of product-preserving functors $\mathbb G \to \textbf{SET}$ is equivalent to the category of groups.
By the way, in general categories that deserve to be thought of as "the same" are not isomorphic.  The natural notion of "the same" for categories is "equivalence".  For example, the $\mathbb G$ has a unique terminal object $1$, whereas in $\textbf{SET}$ any singleton set is terminal.  But a product-preserving functor $\mathbb G \to \textbf{SET}$ picks out some particular terminal object in $\textbf{SET}$, and so in particular there are many more product-preserving functors $\mathbb G \to \textbf{SET}$ than there are set-theoretic groups.
A: I'm not sure whether the following is what you are looking for but:
The theory of groups is a very special kind of theory: It is what is called an Variety of Algebras. This means that the language has only one type of object (an element of the group); has various k-ary maps (in this case, multiplication which is binary, inversion which is unitary and identity which is 0-ary); and all the axioms are of the form "For all $x_1$, $x_2$, ..., $x_r$, the following equality holds..." A model of this theory is called an Algebra in this Variety. This precise example is worked out on the n-Lab page I linked.
There is a notion of a map between two algebras in the same variety. It is simply a map of elements which sends each operation in the first algebra to the corresponding operation in the second. The category of algebras in {Groups} is canonically isomorphic to the category of groups.
Being an elementary map is more restrictive than being a map of  algebras. An elementary map must preserve all first order statements. So, for example, if G is not commutative, an elementary map must take it to a noncommutative group, while a map of algebras need not.
