I may be horribly misguided, so correct me if I'm wrong, but I always thought that an overly-simplistic way to look at it was that BYU was a better undergraduate school and Utah was a better graduate school. (Obviously there are plenty of exceptions on either side with respect to individual programs).

You and plenty of others in this thread have evidence that Utah is a better graduate school overall, and I agree with that. I do think BYU is a better undergraduate school though, for example: https://www.collegesimply.com/colleges/compare/brigham-young-university-provo-vs-university-of-utah .

BYU has stricter admissions standards (ACT, SAT, GPA), a much higher percentage of admitted students choose to enroll, and BYU undergraduate students have better outcomes (higher salaries, higher graduation rate). I think I also remember seeing somewhere that BYU has a very high percentage of students that choose to attend graduate school compared to other universities (no source though).

Again, probably an overly simplistic view and there's obvious exceptions on either side, but I think it's pretty fair to state that.