Category of groups = Category of models of group theory?

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!