The show portrays Joe Gallo being released from prison, which happened in 1971, and then immediately cuts to Tommy Lucchese talking about how he served his time. But Tommy Lucchese died in 1967, so him being in the show at all past the first or second episode is factually incorrect.
The series portrays gangster Crazy Joe Gallo as shaking down Al Ruddy for money during the production of The Godfather. Ruddy contemplates using the money the production earmarked for shooting in Sicily and is only saved by the bell when he learns that Gallo has been gunned down. In reality though Gallo was killed a couple of weeks after The Godfather was released. It's dubious if Gallo was shaking down the production and if he did his murder was not what got Ruddy out of it.
Coppola and Puzo did write the screenplay together but hardly ever worked together in the same place. Coppola preferred to stay in San Francisco while Puzo was in LA. Puzo wrote the first draft which Coppola then edited and revised and added his own touch to the final script.
The Barry Lapidus character did not exist in real life. Instead Lapidus was an amalgam of different executives at Gulf & Western that Ruddy dealt with during production of The Godfather.
Whenever screenplays are shown in the series there are brass fasteners in all three punched holes, but the convention is that only the top and bottom punched holes have them.
Several times throughout the series there are establishing shots of Manhattan that are from the present day and not as the skyline was in the 1970's.
Robert Evans is shown driving a car but according to his own bio and other reports Robert Evans did not know how to drive a car and his assistant Peter Bart often picked him up at home and drove him to the studio.