I typically use Mathematica to prototype ideas. When they pan out, I integrate them with existing code written in c or Python. That Mathematica excels at symbolic manipulation as well as numerical solutions while having excellent visualization tools makes it invaluable.

In particular I love to build up some complicated function of many variables and wrap a Plot[] command in a Manipulate[] command so that I can drag the slider bars around and see how the shape of the function changes. This allow for me to get an intuitive understanding of the problem.