For a math student, does an industrial internship, where you use the applications of the mathematics you learned in courses, give a better preparation for a PhD in Applied Math, or taking more advanced courses? How about taking courses on other topics (not math), which are skill-based or otherwise practical with a lot of applications in applied math, such as high-performance computing, statistical learning, numerical modeling, etc.?
Having been on the grad student selection committee of the departments I was in for the past 4 or 5 years, and having read hundreds of applications, here are the things that really make an application stick out (in roughly this order):
- Having research experience
- Doing well in senior-level and graduate-level courses in the area you are applying for
- Experience applying your knowledge outside an academic setting.
So, if you have the opportunity to do an industrial internship, then you're satisfying the last point and, if you get the internship in the right environment, then also the first point. If that means dropping one of 5 senior/graduate-level math courses (or a non-math course) you were going to take before applying for grad school, then that's a worthwhile trade off. On the other hand, if that means that you're not going to take any advanced courses, then that's not a worthwhile trade off.
First of all, you should narrow down which applied mathematics are you applying. I know people in that area that don't know how to code, and people that use high performance computing. The same applies to almost all non-basic math skills. Applied maths is way too wide to just ask for a general skill.
In my opinion, just general coding knowledge should be fine, you will have time to learn anything you need on the way, that's part of the Ph.D. and, if you follow academia, the rest of your life; you just need to be able to learn new things.
I just would discourage an industry internship if you can use that time for something better (with respect to applied maths).