Too bad “The Omega Glory” didn’t also get in on the game.
That would have been funny.
The difference is that "James T" is actually spoken in "Mudd's Women" (also "The Conscience of the King", "Court Martial", "Tomorrow is Yesterday", "A Taste of Armageddon" and "Errand of Mercy" in the first season) , whereas "James R" only appears visually on the tombstone in "Where No Man..."
I'd guess the real world reason for the discrepancy is that the tombstone was a mistake by the props people that slipped past Justman or Roddenberry. But fan theories that used to go around included that the "R" referred to a private joke or nickname between Mitchell and Kirk, or that the events of "Where No Man..." took place in an alternate reality. (IIRC one of the novels ran with this one.)
Interestingly the "T" middle initial goes back all the way back to Roddenberry's original outline in the early 60s, when the name of the captain was "Robert T. April". Other notable Captains with "T" middle initials include "William T. Riker", "Captain J. T. Esteban" and.... err... I thought there was at least one more, but am drawing a blank.
edit: The other person I was thinking of was "Leland T. Lynch", but he wasn't a captain.