The simplest description of the difference between these two approaches that I have found are on this site who summarise the difference as:
Mechanistic model: a hypothesized relationship between the variables in the data set where the nature of the relationship is specified in terms of the biological processes that are thought to have given rise to the data. The parameters in the mechanistic model all have biological definitions and so they can be measured independently of the data set referenced above.
Phenomenological/Statistical model: a hypothesized relationship between the variables in the data set, where the relationship seeks only to best describe the data.
Could someone exemplify this difference with a short example. I have used and understand the principals of a statistical model (multiple regression etc) but haven't come across any simplistic examples of a mechanistic model and I don't understand what the difference would be in practice.
Edit: I am leaving the question open for a week to see if anyone wants to create a very small 'MWE' of the two types to illustrate what the difference would look like in practice, if not I will accept Memmings answer