Others here must know the details better than I do -- and Hogan never said much of anything about it, as far as I've been able to learn -- but Palmer has written often that Hogan simply never accepted him when he was a young pro, and the coolness continued throughout their encounters on tour.
When Palmer first made a splash at the Masters, Hogan was surprised and said, "Johnny Palmer?" (another pro during the '50s), and he was informed, "No, Arnold Palmer." Hogan had seen enough of Palmer's swing to dismiss him as a guy who'd never win consistently, but Palmer obviously surprised him. Maybe Hogan didn't like to be wrong, and maybe he resented the fact that a guy who had obviously ignored his book had supplanted him as the star of the tour.
Whatever it was, Palmer says Hogan always addressed him as "Fella," never Arnold or Arnie, and Palmer resented that. In a recent Golf Digest interview, Palmer brought it up again, and it was apparent he still hasn't forgiven Hogan.