Continuity mistake: When Lex walks by the military bunker the blinds are opened. When he is inside, they're closed.
Revealing mistake: When the missile is launched, Superman flies to catch it. However, his face has some sort of stocking over the upper part and the man looks nothing like Christopher Reeve. It was either a stunt double, a bad special effect, or a cheap mannequin.
Continuity mistake: When Superman walks to the UN light casts a 1 o'clock shadow on the pavement, yet it disappears a frame later from a different shot.
Continuity mistake: The driver of the SWAT van disappears when the van is standing upright. (01:14:00)
Revealing mistake: When Nuclear Man lands in Metropolis before destroying everything, watch the truck and cab on the left and you'll realise that they are a painting. Not a realistic painting, but a cheap, unfinished one.
Visible crew/equipment: When the cosmonaut is fixing some cables, a crew member's hands holding the camera are reflected on his visor.
Revealing mistake: After Nuclear Man bursts the cab's hood, watch behind and you'll see the wiggling cable that pulls the prop car.
Revealing mistake: The first time we see Clark, he is crossing the crowded street towards the subway entrance. Note that none of the cars are moving except for a cab in the front. If you watch behind Clark you'll see that the street ends in a flat wall, revealing it's a backlot set.
Revealing mistake: When the movie ends and Superman grabs Lenny and Lex's car, they both become black and white coloured and semi transparent.
Revealing mistake: Underneath Lenny's car when it flies away in the quarry there's no pipes, no nothing, just a black plank, revealing the car's a cheap prop.
Continuity mistake: When the firemen are grabbing the hose, the black car on the left turns around 180ยบ and changes model between shots. Also, the yellow cab on the right changes positions and model, notice its headlights changing styles between shots.
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.
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.