Continuity mistake: When Nuclear Man sees the newspaper, it smooths itself between shots.
Other mistake: When Superman flies next to the subway carriage and Lois spots him and smiles, he is flying in the opposite direction.
Continuity mistake: When Clark hears Lois' cries for help he is by the edge of the subway platform. A frame later he is a meter away.
Continuity mistake: Position of thumb changes when Superman carries the Statue, before he is attacked by Nuclear Man.
Revealing mistake: Same footage used when the Statue of Liberty plummets and when it's lifted up by Superman, though reversed.
Continuity mistake: Pavement swaps from dry to wet when the SWAT van is attacked.
Continuity mistake: Before the cop cars drive backwards and crash, all the cops around the cars disappear.
Revealing mistake: Superman's hands move despite him being frozen in a block of ice.
Revealing mistake: Reversed scene when Nuclear Man exits the volcano: note the white smoke climbing upwards into the crater.
Continuity mistake: Nuclear Man's hair becomes shorter when he approaches to read the newspaper.
Visible crew/equipment: When the SWAT van is lifted up there is a crane reflected on SWAT van.
Continuity mistake: During the Metropolis attack, the sky alternates from cloudy to sunny.
Continuity mistake: Nuclear walks towards the newspaper with Lacy's picture and moves his right arm to grab it. A frame later his arm is still, resting by his side.
Continuity mistake: Before Nuclear Man sees Lacy's picture he flies up and produces a big cloud of smoke, which disappears a frame later from a different angle.
Continuity mistake: Lacy flirts with Clark and places her left hand all along his shoulder. A frame later her hand is close to his neck.
Continuity mistake: At the hotel Jeremy walks to meet Mr. Warfield. In the first angle of him entering there's a bellboy walking in front, yet a frame later, from a close-up, the bellboy is now standing still a couple of meters behind.
Visible crew/equipment: When Superman cries "Don't do it, the people" a boom mic is reflected on the glass behind.
Continuity mistake: Nuclear Man kicks Superman and sends him flying hundreds of meters away from the Statue of Liberty towards the sea, but Superman's cape falls down straight from above the torch.
Continuity mistake: When Superman activates his laser vision to cut a mountain he moves his head from right to left, but from his POV the laser beam hasn't appeared yet, plus when it does its position doesn't match Superman's head.
Continuity mistake: Lois opens Clarks's door and stands in the shadows talking to him. The amount of light by the door swaps from gloomy to totally lit depending on the angle, with no order whatsoever.
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.