I found an excellent classification of models in a paper on neurogenesis: Using theoretical models to analyse neural development.

I think this should be illuminating for anyone interested in theoretical, mathematical and/or computational approaches in neuroscience, cognitive science, and biology.

There are several ways in which models of biological processes can be classified. 

Formal or informal models

Informal models are expressed in words or diagrams, whereas formal models — which this Review is concerned with — are described in mathematical equations or computer instructions. Using formal language forces a model to be precise and self-consistent. The process of constructing a formal model can therefore identify inconsistencies, hidden assumptions and missing pieces of experimental data. Formal models allow us to deduce the consequences of the postulated interactions among the components of a given system, and thus to test the plausibility of hypothetical mechanisms. Models can generate new hypotheses and make testable predictions, thereby guiding further experimental research. Equally importantly, models can explain and integrate existing data.

 Phenomenological or mechanistic models 

Most formal models lie on a continuum between two extreme categories: phenomenological and mechanistic. A phenomenological model attempts to replicate the experimental data without requiring the variables, parameters and mathematical relationships in the model to have any direct correspondence in the underlying biology. In a mechanistic model, the mathematical equations directly represent biological elements and their actions. Solving the equations then shows how the system behaves. We understand which processes in the model are mechanistically responsible for the observed behaviour, the variables and parameters have a direct biological meaning and the model lends itself better to testing hypotheses and making predictions. Although mechanistic models are often considered superior, both types of model can be informative. For example, a phenomenological model can be useful as a forerunner to a more mechanistic model in which the variables are given explicit biological interpretations. This is particularly important considering that a complete mechanistic model may be difficult to construct because of the great amount of information it should incorporate. Mechanistic models therefore often focus on exploring the consequences of a selected set of processes, or try to capture the essential aspects of the mechanisms, with a more abstract reference to underlying biological processes. 

Top-down or bottom-up models 

Formal models can be constructed using a top-down or a bottom-up approach. In a top-down approach, a model is created that contains the elements and interactions that enable it to have specific behaviours or properties. In a bottom-up approach, instead of starting with a pre-described, desired behaviour, the properties that arise from the interactions among the elements of the model are investigated. Although it is a strategy and not a type of model, the top-down approach resembles phenomenological modelling because it is generally easier to generate the desired behaviour without all of the elements of the model having a clear biological interpretation. Conversely, the bottom-up approach is related to mechanistic modelling, as it is usual to start with model elements that have a biological meaning. Both approaches have their strengths and weaknesses.

(I removed citation numbers for clarity.)

One point might be relevant here: a model is neither true nor false — ideally it’s an internally consistent mini-world. A theory is the assertion that a model corresponds with reality.