Continuity mistake: When the bellboy picks up Clark with the baggage trolley, he makes a close turn to the right and drives parallel to the elevator. When the angle changes he is in a totally different position.
Continuity mistake: One of the general shots in the subway shows Superman flying slower than usual and inches away from the bystanders. The problem is that their coats and hair don't flutter nor do they notice him passing by or seem nervous in any way. However, the next close shot of their faces show them nervous and in awe at the action.
Continuity mistake: On the moon, nuclear Man grabs a rock with a huge hole underneath and holds it over his head. The angle changes and the rock has changed to a different one with a solid bottom.
Revealing mistake: When Lois and Superman are flying together, they turn green, specially Lois's arms and hands.
Continuity mistake: After Perry White delivers his final speech a woman in a yellow dress goes up the escalator. In the following shot, no one's there.
Continuity mistake: While Nuclear Man is destroying Metropolis, two police cars get moved backwards. The first shot of the cars moving show a lot of people by a shop window. The angle changes to a close shot, with no people there, then back to the previous wide angle, now with the cars moving backwards, and the bunch of people reappear.
Revealing mistake: Outside the UN, a woman in a red coat appears behind the hot dog stand and when the angle changes to the opposite side, in front of her, she's also there.
Continuity mistake: When the Soviets are about to launch the missile in the snowy field, there are two soldiers on the right and a third with a black hat running away. A second later there are three soldiers plus the one with the black hat who's now there.
Continuity mistake: Before the firemen arrive at the truck, there's a lot of smoke around, but half a second later it's all gone.
Continuity mistake: Before Lacy opens the red car's door at the UN, a lot of people walk towards her and stand inches away. A second later they've vanished.
Continuity mistake: While Nuclear Man moves the 2 police cars backwards, the first shot shows the car on the right ahead of the other. A frame later, in a new angle, it is way behind the other car.
Continuity mistake: After Lenny calls Nuclear Man "experiment freako", the stool from the drum disappears.
Plot hole: When Nuclear Man isn't receiving daylight he "dies", yet when he dives into the volcano he manages to come out as if nothing happened.
Continuity mistake: When Lois is walking towards the subway carriage there's a cop controlling the passers by. When Clark walks by two seconds later the cop is gone. The train starts to move and we get a panoramic view of the station and, yes, the cop has definitely vanished.
Continuity mistake: The tornado rips off the farm's roof, but it magically reappears with the house in perfect state after Superman arrives with the girl. He didn't have time to fix it. (Laserdisc and international extended editions).
Revealing mistake: During the tornado, when the little girl cries by the window check the lower corner of the window on the right and you'll see the white string moving the window. (Laserdisc and international extended editions).
Continuity mistake: When Luthor explains his plan to the three baddies the champagne bottle swaps between pointing at him or away. The glass next to the ice cube also moves nearer/further in different shots.
Continuity mistake: When Superman is walking with the kid towards the UN, the sun changes between shots.
Revealing mistake: Right after Lex, disguised as a soldier, orders the launching of the missile, there's a shot of the truck carrying it. The driver is a cheap mannequin that looks very much like a scarecrow.
Continuity mistake: When Superman starts his speech at the UN, in the wide angle his arms are extended, but a frame later they're next to his body.
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.