Is there a difference between a model and a representation? I'm thinking of models in logic here, vs. e.g. group representations.

Is there a difference between a model and a representation? 

Could one not explain both at the same time?
A model gives an interpretation, but this might be viewed as a side effect to the work you do. You have some abstract axioms and you model/realize them with certain objects, which are part of another theory (e.g. the ordered pair concept is modeled via sets and $\in$). I don't see a real difference to e.g. a group representation, when you have some abstract multiplication laws and these come to live via a matrix representation of a specific dimension, say.

Also, 

What kind of realizations do representable functors deal with? 

I mean beyond the realization of groups like above.

Edit, from the comments: Like an example would be to consider integers $\mathbb{Z}$ and build the factor group $\mathbb{Z/2Z}$, this would be representing what is called $\mathbb{Z_2}$ (abstractly defined by the four relations between its two elements). The logic analog would be the sets and the abstract idea of a pair with its characterizing feature.
 A: Well, I mean, they are representations of different kinds of things. But one can think of both as functors out of a given category.
For group representations, a group $G$ may be regarded as a category with one object and morphisms the elements of $G$. A group action of $G$ is then precisely a functor $G \to \text{Set}$; similarly, a linear representation of $G$ is precisely a functor $G \to \text{Vect}$. There are endless variations on this.
For models, a collection of axioms ought to describe category which is roughly speaking the free category containing a model of the axioms. This is easiest to explain for axioms nice enough that they can be described using a Lawvere theory, but I believe this formalism is more general than that. For example, the theory of groups can be described using a Lawvere theory which is described here. Given a Lawvere theory $C$, a model of $C$ is then precisely a product-preserving functor $C \to \text{Set}$. Again there are endless variations on this. See the Wikipedia article on categorical logic. 
In the first example, the representable functor gives you the action of $G$ on itself (exercise), which is roughly speaking the free $G$-action on one element. In the second example, the representable functors for the Lawvere theory of groups give you free groups on finite sets (exercise). 
Edit: Instead of reducing everything to category theory perhaps I should reduce everything to logic. I claim that representations are a special case of models. 
For example, let $G$ be a group. Write down a first-order theory with one unary operation for every element of $G$ and one universally quantified axiom for every entry $g \times h = gh$ in the multiplication table of $G$. Then a model of this theory is precisely a set on which $G$ acts. (A similar but more complicated construction gives linear representations also as a special case of models.) Of course, much of this is redundant: it suffices to specify an operation for every element of a fixed set of generators of $G$ and an axiom for every relation in a fixed presentation relative to the generators. 
