Mechanical engineers make things, industrial engineers make things better. The discipline started in the 1900s mostly concerned with improving manufacturing efficiency and cost. Now it has grown to improve all facets of product quality and business process (physical and digital). It is one of the best disciplines to use applied math. https://en.wikipedia.org/wiki/Industrial_engineering

Industrial engineers work on production processes. You can't make a bridge better without fully "grokking" the physics behind the original design. It isn't exactly amenable to incremental improvement.

https://www.youtube.com/watch?v=g8UePdbDmMw This was one of the keynote speakers at the 2015 IISE National Convention. Nancy Currie was/is works for NASA and has education in Industrial Engineering. She worked extensively investigating the Columbia space shuttle incident. Her involvement shows how industrial engineers participated in making space flight safer / better. It's really not grokking the physics that can make things better. Excellent presentation.