In the scene where the Japanese explorers reach the island, their guide speaks to the island natives in their language. In the English dubbed version of the movie, you can clearly tell the guide has two different voices. (When he talks to the natives in their language, it is not dubbed over.)
American version: the scene of the helicopter pilot screaming "Godzilla!" when he sees the monster emerging out of the ice is kept in the film, even though it is later explained (via new footage shot for the US release) that this is the first time Godzilla has appeared and no one has heard of him before.
During the scenes were Godzilla destroys the train, shots of Godzilla walking towards the train are reused several times, making it look like Godzilla walks to the train, smashes it, then starts walking toward the train all over again.
You can see the eyes of the stuntman playing Kong through the eyes of the Kong suit head, especially near the end when Kong is unconscious, before he gets struck by lightning, in the close-ups of him laying on the ground while Godzilla is hitting him with his tail.
The puppet head used for the closeups of Kong does not look much like the head on the Kong suit, especially the eyes. You can see Kong's eyes, pupil and white part in the close up shots, but only plastic reflecting studio lights on the head of the Kong suit.
When one of the octopus' tentacles grabs a native (stop-motion) and shakes him around, the native itself is a static doll that does not move (its 'skin' is also much darker than the skin of the human natives around it).
When being hauled around by the balloons, Kong's right leg is twisted around painfully and sticks out of him at an odd angle. When it cuts to the suit with an actor inside it, the leg is back to its normal shape.
In the U.S. cut, the same shot of Downtown Ginza is shown every time the film cuts to the lead actors' apartment buildings (you can tell by the street cars and traffic, which proceed the same direction every time).