I’ve always used these terms interchangeably. But somewhere I was reading something that said (something not germane to this thread) was not a “theory” but a “model.”

To me (well, I learned this from my professors), they’re the same. You start with a group of known facts, and construct a “model” (or theory) to show how they might be related.

Fashion Model…Fashion Theory
Model Airplane…Theory Airplane
Number Theory…Number Model

It is quite common for a model of something (such as in the first two examples) to have nothing to do with theory. Conversely, most of the time in science theories are essentually an attempt to model an aspect of reality, but this is not inherent in the concept of a theory.
As the words are used in science, there is also a distinction between a theory and an hypothesis, even though both can be models.

Models are the experiments by which a theory gets tested. A theory generally has a hypothesis that is associated with it. A model is used to test the hypothesis. The closer the hypothesis relates to the theory, the more efficient the model will be when it is employed…this is because the creation of the model become more straightforward than if the theory is ambiguous or confusing.

A change to the model or incorrect data assumptions in the model can yield to erroneous conclusions drawn about the theory.

The short answer, in my humble opinion, is that a model is not synonymous with theory but rather is a subset of theory; a theory tool if you will.

In engineering, a model is a description of a system. For example, in the adaptive filter algorithms used in modems, the phone line system is modeled with a certain mathematical equation. The parameters of this equation are estimated, and the estimted inverse system is applied to cancel the effects of the phone line. The phone line isn’t REALLY described by that model, but it’s close enough to be usable. Knowing how a system really works lets engineers make simplified models that are adequate for whatever they’re designing. However, knowledge of how a system REALLY works isn’t necessary for a useful model. Newton’s model of gravity is very useful even though it doesn’t really model the way gravity works, and he didn’t really understand WHY it worked. To an engineer, the model is the equation, and the theory is the rationale behind the equation (i.e. WHY the equation is correct). Of course, at the most basic level, we don’t know why any of the fundamental physical laws hold, so the difference is moot