For obvious reasons, Utah's admission to the Pac-12 is positive for their program. It means more money, more publicity, better schedules, and better access to the big bowls. However, it also means Utah has entered a new tier of competition. Whereas Utah has two BCS wins, they now compete against schools with multiple national championships. Whereas Utah has in the past lived in one of the more powerful communities in their conference, they are now up against schools that reside in the richest recruiting centers, largest populations, and wealthiest regions in the Western U.S.
Membership means access, and for now, that is understandably exciting for Utah fans, but eventually, they will want more than access - they will want achievement. If they are to be competitive in their new conference, they will need to leverage every advantage they can.
Rivalries are good for programs, because they intensify interest and support for the schools involved, and that passion translates into better media coverage, donations, ticket sales, TV ratings, and - most importantly - recruits. Even when they don't reach the intensity of a "rivalry," in-state games have healthy ramifications for both schools because of the local interest alone. This is why BYU and Utah have scheduled games against teams like USU and Weber State on a regular basis, even during USU's many down years and despite the fact that these schools have disparate conference membership, as BYU and Utah now do.
Utah has put the rivalry on hold for two years, but we have heard just this morning that the Pac-12 and Big Ten have suspended their scheduling arrangement, citing as one of the reasons the fact that "Other Pac-12 schools have regular scheduling agreements with opponents outside the league, such at Utah-BYU." I don't know if Chris Hill communicated his needs to the Pac-12 or how much weight he carries, but I believe he realizes that, in order to sustain his program's growth, he will need every advantage he can get over other Pac 12 schools. Utah's rivalry with BYU may be one of the biggest tools he has.
Eventually, Utah may indeed care less about BYU, but only because they have new competition to worry about. To have a chance against that competition, they may need BYU after all.