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.
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.
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 she 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.
Revealing mistake: When the two soldiers are in the subway tunnel just before Godzilla's eye opens at the end of the tunnel, check out the beams on the flashlights. When the flashlights shine on the walls of the tunnel, they are normal. When they shine FURTHER AWAY onto Godzilla, the light beams get narrower and more laser-like. The circle the light makes on a surface loses the dark circle in the centre when it shines on Godzilla too.
Factual error: The ship shown in the film is a dry-bulk carrier, not a fishing vessel.
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.
Factual error: Multiple US Navy submarines are shown operating in the Hudson. The Hudson River is not deep enough to allow for submarine operations, especially multiple SSBNs (missile submarines which are too large and unwieldy for a shallow river, SSNs or fast attack subs would be used) and having multiple submarines complicates maneuvers and increases the risk for a friendly fire incident.
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: 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".
Continuity mistake: Watch the roads during the chase scenes. They go from soaking wet to dry to wet again scene to scene, in the same street as they turn around.
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."