From what I remember Billy did not like Reggie's ego, and did not like Reggie's style of play. Reggie had a definite label of being a hot dog, and he celebrated home runs. Billy was from the Yankee system which discouraged celebrations. Billy also thought that Reggie was lazy, way too selfish, and a bad fielder.
Also Billy may have been carrying a grudge from the 1972 playoffs. The playoffs had the Tigers, managed by Billy, against the A's, which had Reggie as the biggest star. That series was nasty. In game 2 Billy ordered a pitcher, Lerin LeGrow to throw at A's shortstop Bert Campanaris. Campanaris threw his bat at LeGrow causing a big fight. The series was in the 5th and final game tied 2-2. The game itself was tied 0-0. Reggie stole home against the Tigers, for the only run of the game. Reggie tore a hamstring on the play which caused him to miss the World Series. Billy was upset at not getting to the World Series, and I believe carried a grudge against Reggie for that.