Factual error: When Matthew Broderick teams up with the French Commandos and they are driving into the restricted area - when the French Commando says gum chewing makes them seem more American - they stop at the guard shack. Matthew Broderick leans forward to answer the guard who tells him "I wasn't talking to you, soldier!" In a military vehicle, the highest ranking person sits in the front passenger seat (where Broderick was sitting). The driver is just the driver - usually the lowest ranking member in the vehicle. There is no way a soldier would tell who is probably an officer to be quiet so they could speak to a private driver.
Other mistake: When they are being chased in Madison Square Garden, Nick knocks over a rack of balls and a gumball machine. They show the dinosaurs slipping and sliding. Nick then runs down stairs and joins the others. Shortly, they turn around and see the dinosaurs slipping and sliding on the balls again even though they're now one level lower.
Continuity mistake: In the scene when Godzilla is attacking the three fishing trolleys, the big light posts fall over. When they hit the ship decks, the impact breaks the bulbs and the lights go out. However, a few shots later, as the ships are going under, the fallen lights are back on.
Revealing mistake: After the second attempt to capture Godzilla with the fish, she runs to the sea. You see her jump into the water, yet she makes a very small splash considering her mass.
Other mistake: When they are tracking Godzilla to the second pile of fish in Central Park, the tracking device says that Godzilla is moving with a speed of 175 mph. If you look at the shots before it with the soldier looking over the top of the building, it is impossible for him to walk at that speed.
Factual error: The guys in BDU's (camouflage) have their sleeves rolled up with the lighter colored inside material showing. They are supposed to be Army soldiers. Only the Marines roll their sleeves this way. The rest of us can roll our sleeves up but have to fold the outer layer in a way that the darker camouflage material is still showing.
Factual error: Another CG versus physics problem: When the creature attacks the helicopters from the back, it eats one and the remaining two try to escape. However, the helicopters accelerate at least twice as fast as they would be able to in real life - they start from being still in air and they reach a circa 60kph speed in under two seconds.
Continuity mistake: When Nick has first contact with Godzilla, he signals to the soldiers that they should not shoot. How the soldiers can see from that far and in the heavy rain what his fingers are pointing is one question, the bigger problem is that when we see a close-up of the guy's fingers, there's no rain.
Factual error: The giant footprints in the movie are obviously dug out of the ground. If they were real, the ground would be heavy compressed in a footfrint shape. These look like they were dug with a foot-shaped shovel.
Continuity mistake: They go into the broadcast booth in Madison Square Garden. While Audrey is broadcasting, the French guy moves the vertical blinds to his left to look out the window and sees a dinosaur raise its head. Just after that the view switches to the dinosaur's side and he is now holding the blinds to his right.
Factual error: When Godzilla first attacks New York and then disappears, we are shown the MetLife Building, which has a massive gaping hole in it. The immense structural stress of a gigantic lizard smashing through the building, not to mention most of the center supports being taken with it, should have collapsed the building, yet it still is standing.
Other mistake: When the first fish trap is set up and Nick realises that the manhole covers need to come off, the first covers come off, with the pile of fish in the background about one block away. Nick then runs another 1/2 a block to a block to remove another manhole cover before Godzilla sticks his head up through the road. Taking into account Godzilla's size, we can assume he is about 3 blocks away from the fish. When the shot cuts to the computer screen in the command centre, it indicates that Godzilla is SIX blocks away from the fish.
Factual error: The Apache gunship helicopters seen in the film have three weapons pylons under each wing. In real life, Apaches have only two pylons under each wing. It's possible the makers of the film gave them the extra pylons to make them look "cooler".