Missouri S&T was called University of Missouri - Rolla until a few years ago. Most engineers that I know still call it Rolla. I am an engineer and have several friends and coworkers that went to MST/Rolla and are very happy with their careers.
On paper, RHIT may have better “stats” and rankings but a good student should have a great career (both first job and tools for advancement) with a degree from either school. I would focus more on the environment and fit for your son. Both are STEM-focused schools, but does he want a mid-sized public school or a small private school? If he is unsure and plans to go to graduate school then consider the least expensive one an save the money for graduate school.
FWIW, my daughter was accepted into both and really considered both schools. Ultimately she will be going elsewhere, more for “fit” and specific opportunities but I would have been just as proud if she went to either of these two.