Early in the film during the Osaka night club scenes, stock footage of a stage performance from World War II is used. On the left side of the screen you will see the Japanese red sun banner and next to it is a swastika banner. This would not happen in the post-war years. Some versions of the film have the swastika blurred out with a gray circle.
Godzilla's look radically changes in closeups, when he's portrayed by a hand puppet rather than a suit. This is most apparent when he's seen from the front: the suit has a very wide head with noticeable ears, whereas the puppet's head is more narrow, and the teeth are far more crooked. The eyes also look different.
When the authorities first inspect the dinosaur book, there is a picture of Anguirus in it (put in by the filmmakers). When the book is being read later in the same scene, this picture is gone and replaced by a drawing of two duck-billed dinosaurs (which was the picture really printed in the book).
If Godzilla easily makes his way through demolished, burning cities and withstands military bombardment, why does the fire coming out of the barrels stop him when the military traps him on the island? He could simply walk through it.
When Godzilla breathes radiation on Anguirus, it is clear as the sun that the footage of Anguirus' head has simply been pasted over some footage of buildings being destroyed. He also moves his head around in a very unnatural (and quite goofy-looking) manner.
In the shot where a few people are washed away in a flood in an underground-station, they appear to be transparent, as their images were apparently printed upon the background where the flood was actually filmed.
The effect of the crumbling building's closeup was done via stop-motion, shot frame by frame. However for a frame or two, the entire screen "jumps" to the side and then back, likely because someone accidentally moved the camera before taking a picture.