In the opening scene of the movie, Richard Parker emails a video file on his unplugged laptop from a falling airplane to a hidden location far underground New York City. The scene takes place in 2000, when capabilities of sending video files from two remote locations at such speed were not widely developed and close to impossible.
It is extremely unlikely that Peter - who is incredibly intelligent and has invented the web shooters on his own - would have to watch a YouTube video to teach him that batteries can explode if they are overloaded.
When Aunt May is talking to Peter about doing the washing, she is holding a sandwich with lots of lettuce leaves sticking out of the top. She passes it to Peter and the next shot shows him holding the sandwich but it has no salad leaves of any sort.
After Peter listens to his ear buds while lying on his bed, he sees his father's briefcase in his closet. When he gets up, one of his ear buds falls out, but the next shot when he's walking to the closet door, both ear buds are secured in his ears.
In the Times Square fight sequence between Electro and Spider-Man, Electro sends a bolt of electricity to ram Spider-Man into a police vehicle, denting its door. In another scene seconds after this one, Electro sends another bolt towards a crowd on a staircase, and there, in a slow motion sequence, Spider-Man tosses up the same vehicle to block this bolt, but the vehicle is not damaged, with no broken door.
During the opening chase, when Spider-Man hangs the last container of plutonium from a lamp post with his webbing, the Russian thief easily grabs it, breaking the web. When Spidey immobilises him by webbing his hands, though, he is unable to break the webbing.
When Peter and Gwen hide in a storage closet at the Oscorp building, light beams stream through the holes in the door. For about two seconds the studio light used for this illumination is clearly visible behind Peter's head just outside the door (unless Oscorp has unnecessarily huge spotlights sitting in the middle of its hallways).
When on the phone with Gwen while running late for his graduation, Spider-Man tells her his rapidly changing location is "First and Broadway, Second and Broadway, Third and Broadway." In Manhattan, Broadway does not intersect with First or Second Street.
When Spider-Man is trying to regain control of the plutonium-238 shipment, an onboard computer warns that the plutonium is highly explosive. Plutonium-238, however, is known for not being fissile. Instead, it is used for radioisotope thermoelectric generators because its short half-life makes it release enough energy to stay constantly hot.
Pilots of airplanes on a collision course would all make an avoidance maneuver to the right. In the movie they made a left turn, the controller's heading direction shouldn't make any difference while in sight. However, if the visibility had been good enough to see each other in time, there would have been no risk of collision ; which demonstrates that it was not good enough.
During the opening chase when Spidey hangs the last container of plutonium from a lamppost with his webbing, Aleksei easily grabs it, breaking the web. When Spidey immobilizes him by webbing his hands, he can't break the webbing. This is because he used only just enough strength of webbing to hold the weight of the container.
Gwen reminds Peter to join her family for dim sum at 8 PM in the evening. Dim sum is normally only served in the mornings and at lunch, not as dinner. However, there is no reason for a restaurant which caters for Western diners, to rigorously follow Chinese tradition.
Several times in the movie, when Peter is hallucinating the vision of Captain Stacy right before him, while wearing his Spider-Man mask, the reflection of Captain Stacy is visible in the eyes of the mask. Since Captain Stacy is obviously not there, it wouldn't make sense for him to be visible in the Spider-Man mask lenses.
At the end of the movie, two passenger planes are involved in a near miss. There is no flight control maneuver that has two civil planes flip sideways by 90 degrees, they would just change direction. The move portrayed in the film can only be performed by military strikers. A civil plane attempting a rapid descent would be unable to recover at that altitude.
During the blackout, the hospital and the airport are in the dark and unable to communicate. All such facilities in New York have emergency back-up generators that work off the main power grid, in case of just such an emergency.
In the first battle scene with Electro, and the also the battle scene at the end with The Rhino, there are steel police barricades set up to prevent the citizens from getting in front of harms way. With the possible exception of a parade or funeral that was scheduled to go down those streets, then there would have been no reason for those barricades to be there, as the villains showed up unexpectedly.
When the police and the Rhino are engaged in a pitched gun battle in the street, there are citizens standing nearby behind barricades, including the mother and son who runs out in the Spider-man costume. In reality, everyone would be running in the opposite direction as fast as possible for fear of getting hit. The final shot clearly shows cheering citizens standing behind both the police and the Rhino, oblivious to any gunfire that could restart any minute.