What a good question! My attempt at an answer in one line is that I think that a lot of fields that we consider very important were off shoots of mathematics at some point in the past.
If one goes far enough back I don't think there were many "mathematicians" as we lable them today. I think people were "natural scientists" and mathematics was a powerful tool that they developed for either its own sake or for use in their other studies.
As we gained more knowledge and things became more specialized then things started to branch off. For example, I think you would be hard pressed to find a pure "physicist" or a pure "mathematician" pre 1940s. They did exist, but the two fields of study were much more entwined. With the rise of industrial physics due to the war we comercializad physics and created physics departments (again, they may have existed in the past but not as they do today).
Another more recent example is computer science. A lot of computer science theory was (and still is) developed in mathematics departments. The creation of new computer science programs and departments is a relatively new thing and is again the commercialization of a mathematics off shoot.
I think it would have been near impossible to have predicted the commercialization of physics or computer science in the 1800s, just as it is near impossible for us to predict what will come of it now. If the past is any predictor of the future, though, there are going to be fields of study that don't necessarily exist today that may be born out of mathematical research.