Revealing mistake: Near the end of the movie, Superman descends on the tower, by the elevator room, and his supporting wires are visible.
Continuity mistake: During the volcano eruption the animal on the right (goat or sheep, I don't know) keeps changing from brown colour to white.
Continuity mistake: When Superman throws the rock into the volcano, it flies from right to left into a huge cloud of smoke. In the following angle, it is falling from the blue clear sky, up away from the smoke.
Continuity mistake: When the volcano erupts, a wide shot shows a group of people behind the cop and the priest. In the following shot a woman appears in front of the cop.
Plot hole: Nuclear Man flies through the floor and ceilings of the Metropolis tower without touching anything or anyone; but an outside shot shows the facade exploding for some reason.
Continuity mistake: After the SWAT van is lifted, it tilts forward. A frame later, it's looking sideways.
Continuity mistake: The cab's hood, crumpled by Nuclear Man's power beam, magically smoothes itself a frame later.
Continuity mistake: The rock that hits the prop-astronaut blows his head off. Seconds later the head is back in place.
Audio problem: On the moon, Superman smashes Nuclear Man's head against his knee. The clonk is heard before the hit (even forgetting that sound can't be heard on the moon).
Continuity mistake: Lois visits Clark and takes her coat off, folds it and places it on her arm. A frame later the coat is lying on the couch.
Continuity mistake: When Clark is about to enter the subway train, the people by the door and on the platform appear and disappear all the time: A cop, a man with a hat, a lady in a leather jacket.
Revealing mistake: When Superman flies towards the volcano, the scene of the eruption he is superimposed on is exactly the same one we saw 2 minutes earlier.
Continuity mistake: When Superman throws Nuclear Man down the chimney, the sky is grey and cloudy, but a second later it's pitch black.
Plot hole: Nuclear Man makes his appearance at Lex's house, walks a couple meters and faints "because of lack of sun" according to Lex. This makes no sense, the place is surrounded by windows and there are beams of light everywhere. There are many scenes where he moves around in rooms with the same or less light without suffering the least damage. In this specific scene he simply faints because it's good for the plot.
Continuity mistake: Lois visits Clark at his appartment and a wooden duck, with the head painted green and white and a black coloured body, appears on the table next to her.
Continuity mistake: Outside the UN, the crowd looks up to the sky and watch Superman come flying. When he lands, 2 seconds later, all the stunts on the first row are either gone or standing somewhere else. A kid on the right with dyed hair, for example, is now standing way behind the crowd.
Continuity mistake: On the Great Wall, a woman with a turquoise jacket stumbles and is about to fall off the edge. From a new angle two seconds later, she is three meters behind and stumbling towards the edge again.
Continuity mistake: After the volcano erupts, the boy standing to the right of the girl disappears between shots.
Other mistake: In the rest of Superman movies, whenever he used his eyes to do some trick a beam came out. In this (really cheap) movie the beam is gone.
Revealing mistake: When Superman drops the elevator containing the imprisoned Nuclear Man on the moon, he flies off and lands to straighten the American flag. In this shot you can plainly see the wires helping him land.
Answer: It was made on a very low budget. Golen - Golbus productions bought the rights to Superman. They were mostly known for B-Movies with not so big name stars. It was there attempt to play with the big studios. Plus at least 45 minutes of scenes were cut out, with major subplots.
In addition to budget cuts, they kept shortening the runtime, meaning scenes needed to be cut. The comic book adaptation has the uncut scenes and makes much more sense.