Visible crew/equipment: On the way to the theater Peter intercepts a police officer chasing a couple of criminals. At the end of that scene one of the police cars has a tremendous wreck that swings the car sideways. There is a clear shot of the driver with a black helmet on.
Continuity mistake: Peter has a small horseshoe-shaped scar on his right cheek. In Dr. Octopus's lab, as Octopus is destroying the fusion reactor, they share a meaningful look and the scar has switched cheeks. (00:26:15 - 01:46:15)
Continuity mistake: When Aunt May hugs Peter near her "hero" speech, she hugs him with her head to his left shoulder. The shot changes and you see that she is now on Peter's right shoulder. The shot changes back and you see her on his left shoulder again.
Revealing mistake: In the scene where Peter saves the child (Venus Lam) from the burning building, right after the jump over the floor opening, the child stands up to help Peter we can see she wears knee pads under her pajamas.
Continuity mistake: In the scene where Peter is riding his scooter to Mary Jane's play, after the music fades out you see a black convertible being chased by several police cars. The black convertible is rammed forcibly, and you see many hideous dents on the side of the car. Thirty seconds later, after Peter is done talking to the kids, we see the same car from the same side it got rammed into, but the hideous dents have vanished. (00:24:25)
Other mistake: At the end, when Peter saves Mary Jane from the collapsing structure, he jumps up in the air and swings her to safety. However if you look at the web he is holding onto it is hanging down vertically. What did he attach the web to that was directly above him, the sky? Every bit of the structure around him was/had collapsed and the only thing remotely close to swing from was the cranes. Yet we see that was quite a distance away, so Peter would have had to shoot the web away from himself not above him. (01:47:50)
Continuity mistake: As Peter stops the elevated train with many webs, a tear appears in his suit on the inside left arm during a closeup. Later, when his arms are still outstreched, there are many shots where you can see that the tear has disappeared; for example, a few shots later when he is still trying to stop the train; when the passengers are lifting him over their heads into the train; and many others.
Continuity mistake: On his last pizza delivery before he gets fired, when it first cuts to the receptionist and the room that she is in, the clock is very high up in this shot, above the receptionist's head, then in the following shot when the receptionist is looking in Peter's direction, the clock is much lower. (00:05:35)
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.