Other mistake: At the taxi cab chase at the end, you can notice when the car is a real one and when it's CGI. Note how the real car has scratches and other things on the front of it, but the CGI car, if you look closely, has almost no imperfections, and even if it does have some, it doesn't match the real car one bit. You can see these things when the car drives off his mouth or when you see Godzilla running behind.
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)
Other mistake: It's difficult to spot (it's easier on DVD, but I don't have the timecode). Godzilla has just left Central Park after suspecting the trap with the fish. As she rumbles down the street, hundreds of Apache helicopters swarm in and attack. When they begin firing their cannons at Godzilla, if you look carefully, you'll see the tracer fire from one gunship overlap over another gunship. The cannon fire is heading away from the camera and getting smaller, so it should at least go through the other helicopter and disappear on the other side, but it remains superimposed on the Apache. A little special effects mistake.
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.
Other mistake: Operations on the ground in Manhattan appear to be under the command of "Sgt. O'Neal." Such a sizable operation would never be placed under a sergeant, but would more likely be commanded by an officer, such as a captain or a major. Also, O'Neal is repeatedly called "Sir" by other soldiers. A sergeant would never be called "Sir."