What is the difference between a mechanistic and a statistical predictive model?
Mechanistic model answers the how question. These models are usually biophysically detailed, and designed to be causal. Say you discovered a linear relation between blood pressure drug and heart rate. This would be a statistical model. It doesn't tell you how the two are related biophysically. One could build a detailed model that describes intermediate processes from the drug entering the system, to binding to receptors, and modulating levels of hormones, and acting as a signal to the heart rate modulation system. This model would have more parameters, and if it predicts the phenomena and generalizes well, then it might be a good approximation of the system. Moreover, it generates hypotheses and is more interpretable.

That being said, I do not think the two concepts are mutually exclusive. Some mechanistic/biophysical models can also be statistical at the same time. It's a matter of having enough data and constraints to be able to fit the model or not.