Visible crew/equipment: Lighting screens are reflected on a hotdog seller's glasses when Superman walks to the UN.
Revealing mistake: After Superman sets the Statue back in place, he lands on the ground and the wall behind wobbles, revealing where the movie was retouched to hide the wires.
Continuity mistake: When Nuclear Man steals the Statue of Liberty, its color keeps swapping between blue and green all the time.
Revealing mistake: When Nuclear Man blows the first manhole cover up, you can see the small plug-like device that was in the barrel of the ejector device as it flips over.
Other mistake: As Superman flies past the subway train, it's obvious that the train is sitting still, as the view of the tunnel shown it is stationary.
Continuity mistake: When Superman walks to the UN there's a mounted police man on the right. Shot changes, and he's suddenly on the far left. Shot changes again, and he's now meters ahead, on the left of the UN entrance.
Visible crew/equipment: When Clark watches the subway train leave a stage light is reflected on the carriage.
Continuity mistake: Nuclear man flies over the Russian parade and looks to his right. Then stops and stares in front and shoots fire balls at the tank below. Problem is the parade, tank and everything is taking place behind him (a matte painting). Laserdisc and international extended editions).
Continuity mistake: When Nuclear Man approaches the tornado the clouds and tornado move at normal speed except for the shot of Nuclear in front of the tornado, where it moves in slow motion. Later on, before the family exits the house the tornado moves in slo-mo again. (Laserdisc and international extended editions).
Continuity mistake: When Lois visits sick Clark his fringe changes style from brushed to messy to messier between shots, back and forth.
Continuity mistake: When Lois' car arrives at the UN the rear window swaps between rolled up or down between shots.
Visible crew/equipment: When Nuclear Man's power beams move the Russian missile towards the officers, look below between the missile and the truck and you'll see a crew member moving the missile to achieve the effect. (Laserdisc and international extended editions).
Continuity mistake: When Superman walks towards the UN entrance there's a lot of people between him and the red car. A shot later the people have vanished and him and the kid are standing right next to the car.
Revealing mistake: After (superimposed) Superman stops the tornado, he flies away to the left. Instead, the farmer stares and moves his head to the right. The director forgot to tell him where the image would be placed. (Laserdisc and international extended editions).
Continuity mistake: After flying together, Clark and Lois walk into the apartment. When she says she has "the jeepers" her arm suddenly swaps from lowered to raised.
Audio problem: When Lois and Clark jump off the terrace, she cries his name. In the wide angle she moves her mouth but no sound is heard.
Continuity mistake: In space Superman saves Lacy and flies away with her next to his right side. A shot later she is now on the left.
Continuity mistake: During the Moon fight, Superman is kicked in the head. He puts his right hand on his head, but a shot later his arm is lowered.
Continuity mistake: Clark grabs the crystal with both hands, and slowly starts retrieving his left hand. When the shot changes, his left arm is in a totally different position. (01:12:15)
Other mistake: The part of the great wall of China that Superman fixes where bricks grow has nothing to do with the broken part from before, nor with the wall we see once he fixes it.
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.