The word “mafia” is never mentioned in the film version of The Godfather because the actual mafia demanded it.
At one point the film’s producer sat down with Colombo and asked if he might be allowed to continue the movie without any further intimidation tactics. Joseph Colombo had one request before he would give the okay for the intimidation to stop, and that was that the movie never mention the words “mafia” or “Cosa Nostra”.