At the end, when the camera pans up from the Moon base to view Earth through the crater created by the Gotterdammerung, we see Earth three-quarter covered in shadow; in the next shot, when the camera reverses its position and tracks backward from the moon over Earth we see the sun rise up behind the Moon, effectively creating a lunar eclipse. Were that the case we should have seen Earth as a full disc with no shadow in the first shot.
At the beginning of the movie, the far side of the moon is in daylight, and just before the Klaus and his entourage leave for Earth shortly thereafter, the sun is just setting. However, when the exploratory ship arrives on Earth, the moon is full - that is, the near side is fully lit. Since the length of a day on the moon is actually almost a month, it would take weeks for this change to occur, when the implied length of time in the film is at most a couple of days.
When Dr. Richter first uses the power of the cellphone to power up the Götterdämmerung spaceship, he plugs it to his computer using a cable of his device. Then the phone runs out of battery and Herr Adler takes it in his hand, but it, the cable is nowhere to be seen.
On Earth, Klaus Adler comments a 1971 Volkswagen Type 2 Microbus with the words, "Ah - ein Kleinbus!" While that is a correct word for the vehicle, there is no way he could have known what it was, since the Type 2 wasn't conceived until the late 1940s - and even when it went into production in 1949, it looked very different from the post-1968 model in the movie. There were also no other small vans (Kleinbusse in German) in Germany before 1945; with the word Kleinbus, Germans of that era would likely have identified a small city bus or coach, not a vehicle sized like a passenger car.
The Germans have lived all their lives on the Moon at 1/6th of Earth's gravity. When they come to Earth, they would experience the equivalent of 6G and would barely be able to stand let alone walk normally. Conversely, Washington ought to be clumsy in the Moonbase, particularly when running.
Although there is a presidential election happening, the movie takes place in 2018, a year when there wouldn't be a presidential election. (Presidential elections only happen in years where the last two digits are evenly divisible by 4 - 2012, 2016, 2020.
All weaponry reliant on explosive charges in a vacuum would make no sound and yet when the Washingtons colleague and the lunar module meet their demise on the moon using the luger and RPG, you can hear the gunshot, grenade launch, and the explosion of the LM.
Towards the end, when Renate Richter and Klaus Adler meet again, she makes him point his guns upwards to touch a lamp with broken bulb, so he would get an electric shock. But he was wearing gloves in this scenes and could actually not have had this shock.