(I'd originally thought another possibility was "one costume designer vs. another," but it was Bob Fletcher in both cases.)
More likely "one director and producer vs. another." Roddenberry obviously had his ideas about what the future should look like, and Robert Wise seemed to be touching on a similar aesthetic to his earlier film The Andromeda Strain.
With TWOK, I don't know that Harve Bennett had an opinion one way or the other about the uniforms, but Nick Meyer was obviously in love with that whole Prisoner of Zenda