A formal model is a precise statement of components to be used
and the relationships among them. Formal models are usually
stated via mathematics, often equations.
Formal models can be precisely communicated because they are
well-defined.
Formal models give replicable results. This is the simple meaning
of "mathematical proof".
Formal models are not reality: you must choose the model.
Formal models may not correspond to reality---the prediction will
turn out to be false.
Informal models simply have some lack of precision. Some
relationships may not be stated as equations, or "expert judgment"
will modify the result of the model.