Revealing mistake: It is way too obvious that when Luthor holds the dollar bill, his arm and hand have been replaced by a robotic prop. His arm manages to move without involving all the bones connected to the shoulder and the back.
Revealing mistake: While Superman carries the Statue, there's a close shot of Nuclear Man showing his fingernails. The movie behind is playing the wrong way, and gives the sensation that Nuclear Man is flying backwards.
Audio problem: When Superman is wounded on the neck, and when the italian villagers are saved, the same male falsetto voice is used to add a (crappy and out-of-sync) "Oooh!" and "Aaaah!".
Continuity mistake: When Superman enters the UN hall, a back shot shows the people in the upper bleacher clapping, but a frame later, the front shot shows them sitting still.
Revealing mistake: The close shot of people panicking when the Statue is about to crash against the pavement shows an old woman with a red wool hat and a white scarf. Many scenes later, when Nuclear Man lands on Metropolis, the same lady is seen, wearing the same clothes.
Continuity mistake: While the crowd follows Superman to the UN, the man with the balloons is first seen far away from him, but half a second later he is very close.
Continuity mistake: When Nuclear Man is destroying Metropolis his hair suddenly changes from messy to perfectly brushed.
Continuity mistake: Seen from inside the train-driver's seat, the tunnel is darker and with lights on both sides, but a shot of the tunnel alone shows it brighter and with no lights on the left side. Then it cuts back to the previous angle and everything changes back.
Continuity mistake: Nuclear Man lands in Metropolis and the cab driver on the left swaps several times from having his arm inside the car in the wide angles, to outside the cab's window in the close-ups.
Continuity mistake: When Mrs. Warfield waits for Clark in her office, there's a white sculpture on the table. From the back it's away from a brown object, from the front it's very close to it.
Revealing mistake: When in space, Superman punches Nuclear Man on the back, he turns around and his face has turned blurry and black, but his body keeps the same colour. In the immediate close-up the face's colour is fixed.
Continuity mistake: When Superman leads nuclear man inside the building to find lacy, an old man turns sideways to let Super in. A frame later he is back facing the door.
Continuity mistake: When Superman cools the huge river of lava, a bunch of fresh white and red carnations can be seen inches away. Perhaps they are an ultra-heat-proof breed?
Continuity mistake: A window shop with a red curtain and a TV set explodes and gets totally destroyed, but several seconds later when Nuclear Man plays around with the SWAT van we see it intact.
Visible crew/equipment: Nuclear Man descends close to the statue of liberty to fight Superman. For a brief glimpse his supporting wires are seen.
Revealing mistake: The elevator cables that Superman holds should be greasy, or at least dirty, but they are not, revealing he's merely holding a black rope.
Continuity mistake: When Luthor and Superman exit to the balcony to meet Nuclear Man, Superman gives a step down. A frame later he is up, and several seconds later he steps down again.
Continuity mistake: During the Great Wall of China scene, a tree suddenly appears by the wall just when Superman is leaving.
Revealing mistake: After Nuclear Man rampages Metropolis, Superman cries "Stop!" and a SWAT agent descends from the sky on visible wires.
Continuity mistake: When the baddies are waiting for Lex Luthor to arrive, before he fires them, the position of the cork on the table changes between shots.
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.