Corrected entry: The modified diving helmet used by Edward Judd appears to be a size too small. On multiple occasions his nose bumps against the faceplate and several times his nose is squashed quite flat against the glass.
Corrected entry: When the UN astronauts find the Union Jack on the moon it is tattered and weathered, having been there for over sixty years. However, in reality, this would not occur in an airless environment such as the moon.
Correction: Items left on the moon will deteriorate, but not because of weather (obviously). The damage is caused by other factors, such as the effects of intense sunlight (which can cause fabric to bleach and become brittle), and also because of the constant rain of micro-meteorites (which the Earth is protected from by our atmosphere). Other effects include landslides, thermal stress from rapid temperature changes, etc.
Correction: How is this a movie mistake? Everything was put together by an eccentric inventor, so he simply used what he had available.