Revealing mistake: When Sandman is in the truck of sand and the cop climbs up to catch him, you can see the edge of a platform uncovered. The platform is clearly there so the cop could walk without sinking - he stops right where the platform ends.
Revealing mistake: When Harry and Spidey fly to save MJ from Venom, Harry's face gets replaced by very obvious CGI.
Revealing mistake: Despite the fact that the earlier movies establish that Peter's spider powers give him perfect vision, in numerous scenes throughout this movie (nearly every close up) Tobey Maguire's contact lenses are easily visible.
Revealing mistake: When Venom knocks Spidey off the cab, he then jumps on Spidey's chest. During that moment, Eddie's face is an obvious, cheap CGI (slow motion can help but isn't essential).
Revealing mistake: When Spidey is placing the tubes around Venom, his face is obvious CGI except for some shots of him shouting. Here, a real face (twice as big as it should) has been pasted on the CGI body, looking quite ridiculous.
Revealing mistake: When Peter and Harry are fighting into the Goblin's lair, stunt cables are visible on their backs holding them.
Chosen answer: Answered in the comics: Peter's Spider-Sense tells him if someone is looking when he's about to enter/exit the apartment as Spider-Man.
Captain Defenestrator