Why are both the terms 'structure' and 'model' used in mathematical logic / model theory? Are they just holdovers from different subjects or is there a principled reason for having both?
For clarification, I'm not confused about any actual definitions or usages, just why both terms came to be used; I could, after all, survive perfectly well using exclusively one or the other with little chance of confusion.
A structure is a set with some interpretable symbols(constants, relations and functions) within a fixed language. You do not ask for more from a structure.
However...
A model (of a theory) is a structure which satisfies the axioms of the theory. It makes more "structural sense"...
Maybe an example brings more clarification: Consider the theory of groups. $\mathbb Z$ is a structure in $\mathcal{L}=\{e, \cdot, ^{-1}\}$ but not a model since it is not a group. On the other hand, $\mathbb R- \{0\}$ is an $\mathcal{L}$-structure and further a model as it is indeed a group.
This is what I more or less know within a model-theoretic view. Someone else may give an answer also considering a perspective of universal algebra.
Models are structures, and structures are models. But when we say "model" we mean that there is a particular theory which holds in the structure, and when we say "structure" we are mainly interested in an arbitrary interpretation of the language.
The term 'structure' is a replacement for the earlier term 'system' which was used by various authors (Weber, Hilbert, Dedekind) and meant something along the lines of "a set with added features". The change from 'system' to 'structure' occurred in the 1950's and seems to be owed to Abraham Robinson and Bourbaki. 'Model', on the other hand, appears in Tarski's early works (mid 1930's), and seems to have arisen entirely separately from 'system'. The use of both in modern model theory is, to the best of my knowledge, accidental with only minor intensional differences (as elaborated on in other answers) distinguishing them. There is no principled reason for having both.
Absurdly late to the party but I thought I'd add a comment because this was causing me confusion as well.
In Kunen's book on foundations (The Foundations of Mathematics), a language consists of constants, functions and predicates (and constants are just functions that take $0$ inputs, i.e. they have $0$ arity). To specify a language you just need to say how many functions and predicates of each arity there are. The language of group theory has a constant, a unary function, and a binary function, corresponding to the identity element $e$, the inverse operation, and the group composition. The language of partially ordered sets has no functions but it does have a binary predicate $R(x,y)$ which is "intended" to be read as $x \leq y$.
A structure for a language consists of a non-empty set equipped with constants, functions, and predicates of arity corresponding to those in the language. Each predicate in the language must get a predicate in the structure, each function in the language gets a function in the structure. For the language of group theory, any non-empty domain $D$ along with a distinguished element $e^I \in D$, a unary operation $i^I: D \to D$, and a binary operation $m^I: D \times D \to D$ defines a structure in the language of group theory (here the $I$-superscript is meant to remind you this is just one interpretation of these symbols). The symbols are evocative of the group operations but they could be anything of the correct parity, because a structure for a lexicon/language is not required to obey any axioms.
As an example, if we define $D = \{0,1\}$ and $e^I = 1$, $i^I(0)=i^I(1)=0$, $m^I(x,y)=0$ for $x,y \in \{0,1\}$ then $(D, e^I, i^I, m^I)$ is a structure in the language of groups. However it is clearly not a group (there is no identity element). Similarly, $D = {a,b}$ and $R = \{(a,b), (b,a)\}$ is a structure for the language of partially ordered sets, although it clearly does not satisfy the antisymmetry property. Similarly, for a structure for the language of posets, you simply need a non-empty set and a binary relation on the set.
He uses the term model when considering a set $\Gamma$ of sentences in the language. In this case, $\Gamma$ could be the axioms of group theory (identity, inverses, associativity). For posets, this would be the formulas expressing reflexitiy, antisymmetry, and transitivity. Then a structure in language of group theory may or may not satisfy these formulas (for all relevant elements in the domain). If it does, then we say that the structure is a model for $\Gamma$. A model for the theory of groups will be a group. Similarly, a model for the theory of partially ordered sets is a structure for the language of partially ordered sets (non-empty set equipped with a binary relation) that further satisfies the properties of reflexivity, antisymmetry, and transitivity.
TL;DR: defining a structure one only needs to have the language, whereas to define a model one needs to have a set of sentences in mind. A structure for the language is something for which it makes sense to ask whether a set of sentences in the language in that language hold.
There is no reason for introducing two different terms. Apparently, somebody introduced one term, and somebody else introduced a different term, either because he didn't like the first guy's term, or he hadn't heard of it. Or maybe it was the same guy and he changed his mind, or forgot what he called it before. How would I know, I'm not a historian (nor a mathematician).
The point is, in mathematics there is no official body with the power to decide what the terminology should be. This is different from other sciences, such as astronomy, where some organization claims the power to decide what's a planet. In mathematics, each writer goes his own way, and anarchy prevails. (If you think "model" vs. "structure" is bad, look at the terminology of graph theory.) Eventually, after a few centuries, a consensus is reached. Obviously, model theory (and graph theory) are too young to have reached that point.
One interesting approach to this problem is as follows. Structures contain the domain of discourse and interpretations of the symbols of a given language. Models add a valuation (assigment) function mapping variables to the elements of the domain. So one structure can give rise to infinitely many models. Since we are often not interested in models as such (after all, these are just various rearrangements of values given to the variables), some authors just do not bother with distinguishing models in this sense and talk only about variable assignments/valuations. And so the term 'model' is reused for 'structure' (for historical reasons we have model theory, not structure theory so there's a reason for keeping the name!). So the literature contains all sorts of approaches: models distinct from structures; models considered the same as structures; structures dissapearing altogether and only the notion of model being used in their stead. This used to confuse me no end in my early days but you have to remember that mathematical logic is a relatively young topic and we are witnesses of its growth and change.
