Continuity mistake: When Peter first goes to see Aunt May and tells her he's going to ask her to marry him, Aunt May in the first shot goes to hug Peter on his left shoulder, it cuts away and in the following shot she has hugged Peter on his right shoulder so that the camera can see Peter's face.
Continuity mistake: Before the cop climbs on the truck, he shields himself on the truck's cabin and you can easily notice the tarp is perfectly tied by a dozen strings. When he climbs up, all the knots magically untie and he manages to pull the tarp away. His buddy couldn't have done it because he always appears with the gun in his hand, covering him.
Revealing mistake: When Peter and Harry are fighting into the Goblin's lair, stunt cables are visible on their backs holding them.