Continuity mistake: In the first "Godfather" film, we see Vito Corleone's tombstone, and it says there that he was born on April 29. In this movie, however, Sonny curses the Japanese for bombing Pearl Harbor on Vito's birthday. The attack on Pearl Harbor happened on December 7, 1941.
Other mistake: When young Vito Corleone shoots Fanucci in the chest at the doorway to his apartment, in a continuous shot, a wound/blood bursts from Fanucci's white vest. When Fanucci proceeds to tear his vest open seemingly in disbelief he has been shot, there is no entry-wound, blood or hole in his white shirt directly under where the wound appeared in his vest.
Visible crew/equipment: After the first scene showing the plane starting up its engines and turning around, we are shown another scene of rioters throwing slot machines out of a club. In the bottom centre of the screen you can see a silhouette of what looks to be a part of a jigsaw puzzle piece on the camera. The object must be on the camera because when the camera pans to the right and down, the silhouette stays in the same position in the picture.
Continuity mistake: Shortly after assassins try to kill Michael at his home, he meets privately with Tom Hagen. As Tom enters the room, the table is empty. Then, after they are talking for a few moments, Michael offers Tom a drink from a bottle of courvoisier that has magically appeared.
Continuity mistake: In the scene where Roth's symbolic birthday cake is divided among his partners, we see a shot in which Michael is seen, in long-shot, putting a forkful of cake into his mouth while talking about the rebels. The thing is, it does not sound as if he is talking through a mouthful of cake. What is more, when the scene cuts to closeup, he is not chewing anything, even though it's just seconds later.
Plot hole: During Godfather I, when Michael was plotting the assassinations of the 5 family heads, he mentions to his father that his son Anthony is thee years old, and 'reading the funny papers.' In Godfather II, we begin at Anthony's first communion, and the date given is 1958. Yet, later at the party, Kay mentions to Michael that 7 years ago he promised the family would be completely legitimate. This was before they married and Anthony was born. (This promise was part of Michael's proposal to Kay.) 7 years ago would be 1951, so how could Anthony be 3 years old then?
Continuity mistake: When Senator Geary talks to Michael in his Lake Tahoe office and the camera is on Michael there's an older couple (the woman is wearing a fur jacket) walking by outside. The camera cuts to Geary for a while, and when it cuts back to Michael the same couple is still walking by.
Factual error: When Vito is about to kill Fanucci during a street festival, it is a cold winter's day. Fires are burning in 55 gallon drums along the street and Vito is wearing a heavy overcoat under which he conceals his shotgun. But the scene begins with the shot of a banner that reads "Festa Di San Rocco" which is August 16, i.e. in the summer.
Factual error: When Michael meets with Hyman Roth at his home in Florida the television is on and you can hear the announcer calling a USC/Notre Dame game. The meeting takes place in the late 1950's. The announcer's voice belongs to Tom Kelly, a long time USC announcer, but he didn't start calling their games until the 1960's.
Factual error: When Michael meets Hyman Roth at his home in Florida, they sit in the room with the TV on. You can hear the football game being called by the commentator. The game is a USC/Notre Dame game with the teams called out by the commentator. The meeting takes place in the late '50's before Bautista's overthrow, but the game's commentator is Tom Kelly (very distinctive voice) who didn't begin calling USC games until the early '60's.
Other mistake: When Paolo is shot dead, there's blood dripping down the rocks he's lying on, but there shouldn't be. The rocks are under his shoulder, and he was shot in the back, and a little blood dripped out sideways. And there's no blood on his shirt where his body is touching the rocks.