How is CS/programming used by mathematicians in industry?

Computers can be used to form conjectures such as the Birch and Swinnerton-Dyer Conjecture. You calculate a large number of values and look for patterns to form conjectures.

Another use can be to find counter-examples. One such case was Euler's Sum of Powers conjecture which was disproven by a brute force computer search.

A third use can be to prove theorems by checking a large number, but finite number of cases. One such case is the Four Color Theorem, which has only proven using a computer verified proof which could not be accomplished by humans. Software can be used to prove theorems directly as well.

But I think we should address one other point which is that the Curry-Howard Correspondence shows that all proofs are programs and all programs are proofs. This means the idea that math being used for computer science and vice-versa might not be a meaningful question in the first place. They're both the same thing, so you're always using both of them. They're entangled with each other inherently.