Mathematical modelling
I will talk about mathematical modelling in the hard sciences at first, in the soft sciences thereafter, with emphasis on essential differences between the two. It may be useful to start with semantics: the "hard" sciences are, in the present context, those (such as the physical sciences) for which mathematics is a natural tool. Some of those mathematics were in fact created to allow for a description of systems in the hard sciences: indeed, Newton was successful in his mathematical description of the motion of celestial bodies because he had at his disposal the infinitesimal calculus that he had just "invented". Calculus may thus, with good reason, be called "the" natural tool for the description of physical dynamical systems (the broader class of objects to which the same mathematics apply). By contrast, the "soft" sciences are those for which such a natural correspondence does not exist, where mathematical models are often artifacts, intended only to describe approximately certain aspects of a system under study (such as in biology, economics, some of the environmental sciences and the like), where they may contain inherent uncertainty, or where they may describe the average of a collection of things, each member of the collection having its own hidden peculiarities. While the distinction between hard and soft is not always clear (it may be argued that some systems fall somewhere in an in between zone), the two extremes of this spectrum are undeniably well defined. Of equal significance is the fact that the difference between the two is not just a difference of tools and techniques. It has also an intellectual component, in that those scientists who do mathematical modelling in the soft sciences are compelled, much more so than in the hard sciences, to play a personal role in making choices in the formulation of models at the beginning, in interpreting the results obtained at the end, in explaining what they mean.
About "mathematical modelling", also in the realm of semantics: It is only in the past few decades that this name has gained notoriety, in particular since electronic computers became a