Factual error: Towards the end of the film, Spiderman takes his web and uses it to connect to the different power conduits that Electro blew open, ties them together, and power the city back up. Electricity uses several different wires, and crossing them would cause things to blow up or short out. What should have happened when Gwen threw the switch was another power failure, instead of the city's power being restored.
Visible crew/equipment: In the scene when Spider-Man is chasing a truck stealing OsCorp's plutonium vials, the truck, driven by Aleksei Sytsevich, crushes one of the police cars. The scene cuts to the left side of the truck crushing the police car. A moment later, it cuts to another angle. Here you can see the camera equipment on the left side of the truck.
Other mistake: The main events of the film presumably take place in May, since we see Peter and Gwen graduate in the beginning. After Gwen's death, Peter is shown mourning through autumn and winter-like weather. However, near the end of the film, a news reporter says it's been only five months since Spider-Man disappeared. This would put the last fight with Rhino in October, and mean it snowed in the middle of summer.
Continuity mistake: In the scene where the power is out and two planes are on a collision course with each other, the microphone of one of the air traffic controllers changes from a correct position facing his mouth, to an incorrect position hanging way down and away from his mouth, between shots.