TL:DR to answer your question - Utah has better grad programs in:

• Physician's Assistant
• Nursing-Midwifery
• Physical Therapy
• Pharmacy
• Nursing/Doctoral
• Audiology
• Medical/Primary Care
• Chemistry
• Speech-Language Pathology
• Mathematics
• Med/Research
• Earth Sciences
• Occupational Therapy
• Computer Science
• Health Care Management
• Social Work
• Clinical Psychology
• Engineering
• Physics
• Biology
• English
• Psychology
• Sociology
• History
• Econ
• Political Science
• Public Health

BYU has stronger programs in:

• Law
• Business
• Education
• Nursing/Masters
• Public Affairs

BYU is clearly the best for undergraduate. For example, for 2022, BYU ranked (by USNews) as the #79 university in the US. And that is absolutely higher than any other in Utah, but still ever so slightly subpar (or sub-median if you want to be more specific) academically among P5 schools. And the gap between Utah (#99) and BYU is substantially smaller than the schools we want to claim we are similar to (Virginia, Notre Dame, UCLA etc).

Looking at graduate schools, it's honestly more complicated to rate. Utah produces much more research, BYU has very strong business/law programs (although we might overrate those as well - we're #29 for law and #31 for business, both of which are behind a school like Arizona St (#25 law, #30 business) that we mock as academically pitiful). Even in those strongest areas of ours we're not that far ahead of Utah (#43 law, #44 business) and that ignores their strong areas. That said, I did include a grad school rundown at the bottom.

I still believe that BYU is a clearly better undergraduate academic institution than Utah (there's a reason I went there and it wasn't just the scholarship).  But I do think we overrate the academics and we assign so much of our self-worth to being better academically that some BYU fans or grads become offended when people don't view other schools as lower-class than BYU.  Which is a pretty messed up superior attitude to have and I think that rubs some people the wrong way. Overall (according to US News at least), there's not much difference (academically, obviously there are major other differences) between BYU and Arizona St or between BYU and Utah. But we'd be offended highly to have anyone say something like that.

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So here's a brief rundown of BYU vs Utah (again according to USNews)

BYU
Undergraduate Ranking: #79
Top 10 Grad Programs: 0
Top 25 Grad Programs: 0
Top 50 Grad Programs: 3 (Law #29, Business #31, Public Affairs #48)
Top 75 Grad Programs: 5 (Above + Speech-Language Pathology #63, Education #75)
Top 100 Grad Programs: 9 (Above + Nursing/Masters #81, Mathematics #86, Computer Science #91, Chemistry #96)

Utah
Undergraduate Ranking: #99
Top 10 Grad Programs: 1 (Physician's Assistant #4)
Top 25 Grad Programs: 5 (Above + Nursing-Midwifery #11, Physical Therapy #13, Pharmacy #14, Nursing/Doctoral #23)
Top 50 Grad Programs: 17 (Above + Audiology #29, Med Schools/Primary Care #32, Chemistry #32, Speech-Language Pathology #32, Mathematics #34, Med School/Research #41, Earth Sciences #41, Occupational Therapy #42, Computer Science #43, Law #43, Health Care Management #43, Social Work #44, Business #44)
Top 75 Grad Programs: 23 (Above + Clinical Psychology #56, Engineering #58, Physics #61, Biological Sciences #62, English #69, Psychology #75 )
Top 100 Grad Programs:  30 (Above + Sociology #79, History #81, Economics #83, Public Affairs #83, Education #85, Political Science #89, Public Health #90 )

Even taking out any medical/health related things, that leaves BYU with 3 top-50 programs, 4 top-75, and 7 top-100.  But Utah has 7 top-50, 11 top-75 and 17 top-100.