Plot hole: Dr Octo builds a new, bigger machine and it starts to pull the building in on itself, but it's so strong it also manages to pull all the metal from the streets as well, including cars. When Spiderman saves MJ, the metal cranes have somehow managed not to be pulled in or at least bent by this massive machine that made a building collapse on itself.
Character mistake: In the start of the film, the sticker on Peter's head for his work misspells the word Bleecker Street on it.
Deliberate mistake: Perhaps to avoid getting an R rating or parental complaints notice how in the operating room massacre none of the doctors appear to bleed nor do Doc Ock's tentacles have blood on them.
Suggested correction: The problem is in both versions of the film, basically all of the deaths either happen off-camera, or are things like people being shocked to death or killed by being tossed around and presumably having their backs/necks broken. Most of the deaths would be bloodless or near-bloodless anyways, and the camera simply doesn't linger on the bodies enough to show any blood. This is also why the tentacles aren't really bloody. Sure, it was probably ratings-related, but it isn't technically a mistake.
Other mistake: During the hospital scene where the robotic arms kill all the operating room staff, we see 4 different camera angles on the last doctor to be killed. All the arms have cameras in them, and show his face screaming at 4 different angles. Yet it should only be 3 as one arm has its claw wrapped around his neck. And it doesn't release from him, as it can be seen still wrapped around his neck by the other 3 cameras in each claw.
Visible crew/equipment: The camera and a boom mic are visible in the police car lights in the scene directly following the first accident with Doc Ock.
Continuity mistake: When Spidey faints after stopping the train he is carried away by the passengers. One of them raises her right arm to help. A shot later her arm is lowered, and she is raising it again. (01:36:00)
Continuity mistake: Right before Spidey stops the train there's a lot of people inside next to him. Then a wide shot later they've vanished only to reappear a shot later.
Continuity mistake: Spidey throws a huge iron minute hand at Ock which destroys an enormous chunk of concrete and makes loads of debris fall downward. However, a shot later there's a view of the street and the train track and not only is the debris gone from the road but the traffic and people act as if nothing had happened.
Continuity mistake: After the car slams the cafe, Peter watches the street with people running around. The reflection on his lenses has nothing to do with what the opposite angle shows.
Continuity mistake: When Peter/Spider-Man has his first web failure and falls on the pipes on the building's roof, he keeps his hands free in one shot and then his right hand is put on his chest in the next.
Visible crew/equipment: After he watches Mary Jane exiting from the theatre with her boyfriend, Peter is running in the alley to put on the costume. On the wet ground you can see the traces of the tires belonging to the camera car used to shoot the scene.
Continuity mistake: Kids buying fruit suddenly appear behind Peter when he arrives at the pizza place for the first time.
Continuity mistake: Peter back flips off his moped to avoid the car racing at him. There were cop cars and other vehicles behind that car when Peter jumps, but when he lands only one cop car goes past in the lane next to him and the lane he's standing in is then clear of traffic as he talks to some kids.
Continuity mistake: When Doc Ock attacks Peter in a cafe with MJ, he throws Peter at the wall. When Peter stands up of the wall and walks outside you can see everyone panicking and screaming. In the shot when Peter takes off his glasses and the glasses hit the floor, you can see in the background that everything is quiet and no-one is panicking, as if nothing happened. (01:32:50)
Continuity mistake: When Peter first sees the burning building, there is a far away shot and you see the guy with dreadlocks run to a woman. Then it cuts back to Peter as he tries to become Spiderman then remembers he can't. Next shot has the same shot, only this time it's a close up of the man with dreadlocks running to the woman.
Continuity mistake: After the bank scene where Spidey saves aunt May from Doc Ock, while they are on the street the sun changes between shots.
Revealing mistake: When Peter returns to his old life and walks on the notes of "Raindrops Keep Fallin' on My Head", at a certain point he stumbles and behind him some people look at the camera.
Suggested correction: They are looking in the direction of the camera. It doesn't necessarily mean they are looking at the camera. They could just as well be looking at Peter since he stumbled and fell, which is a pretty normal thing for people to do when someone falls down.
Actually people behind him keeps watching even when Peter restarts to walk normally. I think if they were really watching just him, they would have looked away immediately as he got to his feet.
