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.
Plot hole: Harry tells Doc Ock that in order to find Spider-Man he must find Peter first. Doc Ock finds Peter with Mary Jane in the cafe and throws a car through the window straight at them, then later throws him against a brick wall. Any normal man would've been killed instantly, and Doc Ock doesn't yet know that Peter is Spider-Man. Given that Peter is his only lead on Spider-Man, it makes no sense that Doc Ock would effectively try to kill him.
Continuity mistake: During the train scene, Spider-Man's mask had gone partially black. We also see it when Spidey puts his mask back on. Yet when Doc brings him to Harry, we don't even see a patch of darkness on his mask.
Factual error: Considering the brightness of the fusion process, Dr. Octavius has to wear special goggles to be able to see it. Yet no one else in the room is wearing such goggles or seem hurt by watching the whole process, just as at the end of the movie. When welding something, no one can look at the arc that's created, as it would hurt his eyes and burn his retina; presumably, the fusion process would be brighter and more powerful than that, and so should have some kind of damaging effect on everyone's eyesight (except Spider Man's, maybe).
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: When Dr. Octopus is robbing the bank, NYPD squad car 4619 arrives on the scene. However, squad car 4619 was badly damaged during the pizza delivery car chase earlier in the movie - an awfully quick fix.
Continuity mistake: When Peter misses the class at the university, the way he holds his books/folders changes when the teacher talks to him.
Continuity mistake: After Spider-Man recovers from the train incident, Doc Ock smashes in the other end of the train and throws the door on the floor in front of him. The camera cuts to another shot, and when it returns the door is no longer on the floor, nor is it there for any subsequent shot.
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.
Continuity mistake: When Peter arrives at his aunt's home at the beginning of the movie, it's night. He talks to Harry in the kitchen a few minutes later, and look at the purple balloon by Peter's head, it reflects a window with lots of light coming through it.
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.
Factual error: Dr. Octavius says his fusion relies on tritium and that there is only 25 pounds of the substance in the world. In reality, tritium is a good deal more common than that. From this article (http://www.epa.gov/radiation/radionuclides/tritium.htm): "It is used in various self-luminescent devices, such as exit signs in buildings, aircraft dials, gauges, luminous paints, and wristwatches. Tritium is also used in life science research, and in studies investigating the safety of potential new drugs." There's also a large region of water near Antarctica which is rich in tritium.
Continuity mistake: When Aunt May is falling off the building, her purse is on her right arm. The next shot shows the purse on her left arm.
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.
Visible crew/equipment: At the very last scene of Mary Jane looking over the city in Peter's apartment, there is a reflection in the window where the camera lens and cameraman are partially visible.
Continuity mistake: Regarding the scene where Doc Ock has delivered Spider-Man to Harry's mansion, when Harry pulls off Spider-Man's mask, there is a very obvious and bad splice. The "mask on" Spidey's head is in a different position and further away than the "mask off" Spidey's head.
Continuity mistake: During the final conversation between Spider-Man and Doc Ock, the rips in Spider-Man's suit keep changing. For instance, there is a tear on his right shoulder; for most of the scene, there is a single piece of black webbing left holding the rip together, but when Doc Ock grabs Spider-Man's arm, the rip now has two pieces of black webbing. Then it goes back to one.
Continuity mistake: In the scene with Peter's "I'm back!" jump, It shows a shot from above and you can see his solid outlined shadow in front of him. However, in the very next shot it shows you a close up of Peter walking, and there is no longer a shadow there.
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.
Deliberate mistake: When Spider-Man shoots a single web out of each arm to stop the train, in the shots of his reactions, you can see that he is holding many webs in each hand. Then when he shoots multiple webs to try and stop the train, he has the same amount of different webs as before. Obviously they didn't have footage of him just holding a single web, and used shots from later in the sequence.