Someone can argue BYU has a stronger tradition/pedigree because they were very good in the 80s, but at this point, that's literal lifetimes ago for the kids who are choosing schools now. Those kids were born in 2004 or 2005; their parents probably aren't even old enough to have been following BYU in their heyday. At this point, it's as relevant as Utah winning a bunch of game over BYU in the 40s, 50s and 60s.

If anything, I'd think BYU would be slightly worse off - the LDS influence currently gives them a better chance at players that wouldn't normally consider a BYU-level program, which gives them advantages over other G5. At a P5, they'd still have the downside of that though, kids not wanting to go to a "Mormon school".

All things considered, I'd guess that BYU would have a rough acclimation, and probably be competitive with Utah, maybe slightly behind in aggregate over multiple seasons, but very close overall.