Factual error: A helpful automated voice from Oscorp informs the robbers that "Plutonium-838 is a radioactive material and is highly explosive." No it's not. It is radioactive obviously, but it is not some volatile explosive. (00:09:10)
Plot hole: The crux of the plot (one of the many plotlines, that is) is that Harry Osborn hates Spider-man because he refused to help him when he is quickly dying of the same genetic disease that killed his father. You can see the problem in this sentence already; he is a teenager, but he's dying super-quickly (he already has a large ulcer on his neck) of a disease that didn't bother him with severe symptoms before and that took away his father at 63. No reason is given of this sudden outburst in Harry's case, in fact nobody even acknowledges that such a difference exists.
Plot hole: Harry learned about Max Dillon watching the Ravencroft video that lasted just a few seconds; from that he can infer the hatred for Spider-man, but not much else, and he's instantly locked off his Oscorp account, and building, after that.However, when he meets Max, he also knows that he is the one who submitted the design for the Oscorp power grid, which is information he shouldn't have since he had no time nor opportunity to investigate further - plus, all data about Max has been erased, and he never got proper credit for that to begin with. (01:28:50 - 01:36:30)
Other mistake: When Peter jaywalks to meet Gwen just after Max is reborn as Electro, a car hits him, breaking its side mirror against his amazing body. Peter does not even flinch for the hit and the mirror shatters sending sparks all over the place. As sturdy as Spider-man is, he's not made of actual solid steel. (00:45:45)
Other mistake: Spider-Man learns about the plutonium heist because he overhears it from a helicopter as he jumps by it. Admitting that Spider-Man has enhanced hearing that he doesn't show to that extent in the rest of the movie, he also manages to completely block out the noise to pick up the person speaking inside it, and is unaffected by the changing distance (they both are in motion in different patterns).
Continuity mistake: When the Statue of Liberty tells Spidey "Nice outfit!" the people in the background are in different positions at the cut. (00:21:20)
Continuity mistake: When the kid dressed like Spidey slips through the barricade, look at his mother in close-up. To her left (right of the screen) there's a guy with white hair that breaks into a sound reminiscent of the famed Aro's Laugh. In the next shot, the guy is in a different position, to the right of the previous position, closer to the asian guy. (02:10:25)
Continuity mistake: Spider-Man leaves the "I love you" message to Gwen on Brooklyn Bridge, and on the tower of the bridge you can see the American flag. When the bridge is in the background later, the flag is gone. (01:45:15)
Continuity mistake: When Electro pierces the chest of the head of the security, the guy falls at a certain distance from the chair and table of the office, but when Harry saunters off to his body to grab his gun, the dead guy is right by the furniture. (01:40:05)
Revealing mistake: Max is having a delusional monologue at the mirror with his imaginary friend. You can notice that the 'gap' between his teeth is represented by a piece of black tape. (00:24:30)
Continuity mistake: When Peter pulls the boot of the costume under the cover to prevent May from seeing it, you can get a good look at the stack of books he's going to trip into a few seconds later. In fact, you can still see even as he trips, that the stack includes the famous leather bag from his father. He doesn't knock it to the floor when he trips, but the next shot just shows it on the floor anyway. (00:23:25)
Continuity mistake: The little bullies are pushing the wind turbine kid against the fence; they literally have their fingers inside the links; they look up and notice Spidey, barely move very slowly, but a moment later they are much further back in the alley giving him a solid few meters of space where he lands. (00:20:35)
Continuity mistake: Coming out of the Chinese restaurant, Emma Stone is closing her coat, and you can see in the view from behind that she even completes the movements to properly tie the belt. The scene continues, the belt is perfectly fitting and closed, but the last shot of the scene shows Gwen Stacy in front of the door with the coat open and the belt loose. (00:17:35)
Continuity mistake: When Peter is kissing Gwen by the podium, Garfield's right hand is on Emma Stone's cheek, but in the reverse angle his hand is on the middle of her back. (00:14:35)
Continuity mistake: The very first time we see Max Dillon, he's carrying literal blueprints around. He mumbles "There's more room on the sidewalk"; notice behind him just after he says "Excuse me", a man with white hair and beard, smartly dressed with a handkerchief in his breast pocket. The shot changes and the same extra is coming from the opposite direction in front of Jamie Foxx. (00:09:15)
Continuity mistake: In the opening scene, Richard Parker hears his son Peter calling his name, and he rushes to check on him, opening the door to his study who has been ransacked by someone. That's the same scene as the first movie, only it is absolutely not the same; not only the position of the items in the room is different (and so also the lighting, since lamps were amongst those objects) but Campbell Scott opens the door using both hands instead of just his left (check out the same timecode of the first movie). (00:02:10)
Revealing mistake: During the "Gone Gone Gone" montage where Peter recaps the parents subplot by plastering his room with various junk about them, he sticks a particular newspaper clipping using tape. The article is an obvious movie prop fake; look at the right of the photo and you can see the paragraph begins with not even a proper sentence; "Cell regeneration, and long time partner and friend to Norman Osborn, went missing", which is the copy and paste from the beginning of the article save the first line. Copy-paste can also easily be spotted from the leftmost to the rightmost column. (01:01:50)
Other mistake: Peter grabs with his web at super-high speed a mug full of scorching hot liquid that he uses to slapstick his way out of the Oscorp security stooges; somehow the cup clashed against his hand staying perfectly filled and unperturbed, but in the next shot the slightest bump spills coffee everywhere. (01:11:20)
Visible crew/equipment: We get a good view of the Oscorp corridors, in particular the one outside the closet where Gwen and Peter hide. They kiss and there's a lovely light seeping through the door's window; when they start making out and also quickly when they open the door you can actually see the big spotlight that emits those warm rays. (01:10:49)
Continuity mistake: When Gwen nearly falls off her own flats trying to avoid Oscorp's cartoon security, Peter grabs her. She clings to her and her mug hand is against his arm. She says "Hi!" in the next shot and her arm is lowered. (01:09:20)
Answer: Peter could have contacted the police anonymously.
MasterOfAll