Factual error: The Apaches in the film are depicted with dual machine guns on each side of the cockpit, mounted directly on the windows. In real life, Apaches cannot mount such cannons, and instead use an M230 Chaingun in a turret underneath the helicopter, which the film Apaches have, but for some reason do not use. (00:49:25)
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 her to walk at that speed. (01:18:15)
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: F-18 Hornets deploy Harpoon anti-ship missiles to destroy Madison Square Garden. The Harpoon is an anti-ship missile, designed to punch holes in ships. It is not designed, nor capable of levelling buildings. A Harpoon also works best over water; firing over a crowded city confuses its radar seeker head and leads to tracking issues.
Factual error: A sergeant is neither called sir nor is saluted. Sergeant O'Neal is called sir twice and saluted twice. The first time is shortly before they dump the fish for bait the first time. Someone runs up to O'Neal, says sir and salutes him. He says something about ten minutes then salutes him again and runs away. The second time is when O'Neal is on the roof and looks over the parapet and sees Zilla walking past. Right after that someone comes up to him and says "Sir, Godzilla has been spotted in section five."
Revealing mistake: When Godzilla attacks the fish market, the creature picks up the seafood/fish truck with the driver still inside. As he's being knocked out of the cab, you can see dozens of boxes set up where the truck was once parked for him to land on. (00:26:00)