For the players, if they want to have input on design, features, or materials that they'll play and practice in, they'll be able to walk over to the UTAD and make those requests. I don't know if that's been the case under Nike.
Since this thread is in the Lady Vols Forum, there is one aspect I can think of that might be relevant to female players and recruits:
IIRC, Nike at some point came out in support of transgendered men playing in women's college sports. Apologies if that's not accurate. But if it is, I'm guessing that most female athletes might not appreciate that stance, and it could be made another selling point to some recruits in favor of an Adidas school.
(And I don't know... Adidas might have expressed similar support under the DEI pressure that BlackRock's CEO Larry Fink was exerting over corporations a few years back.)