This is a classic question that has been debated by physical scientists for decades if not centuries. Heisenberg (that most uncertain of physicists) insisted that every quantity appearing in a model should be intrinsically measurable. One of Einstein's most fundamental contributions was the recognition that physical theory is best written in terms of what we actually do in the physical world. This operationist view was adopted and extended by Percy Bridgman in what I think is the best book on this subject, The Nature of Physical Theory, Wiley Science Editions, New York, 1936. Bridgman, however, required only that some variables in the mathematical theory correspond to measurable quantities.
In compartmental analysis, we see a spectrum of models being developed "to account for" or "to explain" the dynamics of real physical-chemical systems. This spectrum runs from descriptive to mechanistic.
Descriptive models aim to find some functional form that "fits" the system's response, or to reproduce a measured output given the corresponding input. A fit to a sum of exponentials or to a power series or to a Fourier series can reproduce a system's dynamic response in exquisite detail, but few scientists would assign any physical meaning to the parameters of such a fit. A transfer function, of the kind used frequently in engineering disciplines, resides a little further along the spectrum toward mechanistic. It has the advantage that responses to new inputs can often be calculated, at least if the underlying system is linear and its properties are not altered by the new circumstances.
Mechanistic models lie at the opposite pole of my spectrum. Each of the differential equations in a mechanistic model describes the determinants of change in a particular physical-chemical quantity. If not measurable by current technology, it is usually conceivable that a method could be developed to measure all of these quantities.
Many published models fall somewhere between these two extremes. Some of the compartments correspond to real or hypothetical physical quantities. Others are introduced solely to account for particular features of the data.
If such compartments are introduced without a hypothesized link to the physical world, their presence cannot be the subject of an experimental test. A model containing compartments not attached to the real world is therefore weaker than a model whose compartments are all related to real or hypothesized measurable quantities. It's weaker because it does not make as many predictions that may one day be compared to experimental observations.
Another way of saying this is to contrast a mechanistic compartmental model with a sum of exponentials. Any thoughtful scientist will think of many ways to test a given mechanistic model, but there is no way to test a sum of exponentials. Each compartment added to a model without at least a hypothesized link to reality is, in effect, adding one term to a sum-of-exponentials description of a particular feature of the data. Without doubt, such compartments improve the fit of the data, but they do not, in my view, advance science.
My answer, then, is that a model whose compartments and parameters all correspond to real or hypothesized physical quantities or processes is always stronger than one that includes purely mathematical constructs. It is stronger because it is more mechanistic, and less purely descriptive; it is stronger because all of its component parts are subject to direct experimental test.