Does letting the Conductor or DJ Grooves win change anything?

The changes are (mostly) purely cosmetic.

The following assumes that you have completed Chapter 2 (including the boss fight), and will not use spoilers.

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• Award Ceremony: the winner will thank you and give you the time piece.
• Boss fight: the winner will fight against you, the loser will support you.
  • Video of DJ Grooves fight
  • Video of Conductor fight

Aside from that, there is not much difference. Both the Conductor and DJ Grooves have the same lines (except the Conductor calls you "lassy" and DJ Grooves uses "darling"). The boss fight remains identical, and even the winner's reason to fight you is exactly the same.