Continuity mistake: In Dr. Hanson's lab, the model brain on the table changes positions between shots.
Continuity mistake: Billy opens the Gremlin's cage and leans forward to look at it. A frame later he is standing away from the box.
Continuity mistake: After Pete meets Gizmo, Billy grabs Gizmo by his head. A frame later he grabs him by his tummy.
Continuity mistake: Before the Gremlins attack the old lady, Gizmo is in the police office with a green string over his head. The string keeps changing positions between angles.
Continuity mistake: While Mrs. Deagle complaints at Billy for the broken snowman, her head is first to the left of the snowman, then to the right, without her having moved.
Continuity mistake: While Billy is trying to start his car, the wide shots show the door totally covered in snow, but in the close-ups there are uncovered parts.
Revealing mistake: In the scene with the malfunctioning juicer, you can see the actor who plays Billy is about to laugh right before all the juice explodes out of the machine. As that is happening, you can see he has broken out into a full laugh. True, he might be showing some mirth over another one of his father's malfunctioning inventions, but he was about to break out into laughter before anything happens, and his laughter is inconsistent with his frustrated demeanor when Pete walks into the kitchen. (00:22:40)
Deliberate mistake: At the bar many of the gremlins are drinking beer and getting wet from it but nothing happens.
Revealing mistake: Gizmo flies off a snow shovel on his car and it blows up. There is a noticeable edit here where the effect happens. It's likely the car didn't blow on the first take so they spliced two scenes together.
Other mistake: One of the evil Gremlins is messing with a stoplight by shorting out wires. The result is a very bright spark. Bright light is supposed to have a negative effect on the Gremlins - when we see them hit by the headlight beams in the bar they scream.
Revealing mistake: When a Gremlin shoots Kate, the glass is already shattered before the bullet hits it. (01:17:15)
Continuity mistake: When the movie ends, the sling around Billy's neck is either over the collar or under, depending on the shot. (01:38:35)
Continuity mistake: In the kitchen, Billy's mom shields her face with a table. A frame later it's at waist level. (00:54:50)
Continuity mistake: When Bill is in the bar drawing a cartoon of Mrs Deagle, the amount of popcorn on the table increases between shots.
Continuity mistake: After Gizmo poops, there's a close-up on Billy's stunned face, perfectly lit. A frame later, from a different angle, the face is in shadows.
Continuity mistake: Both the scene of Billy, Pete, and the science teacher looking at the gremlin cocoon, and later when Pete is in class watching the science film, occur on the same day, yet Pete is wearing different clothes.
Continuity mistake: When Billy smashes the Candy store window the amount of stuff behind the glass changes between shots. (01:27:50)
Answer: The original film was released before the PG-13 rating existed (it was one of the films responsible for its creation). Producers decided to remove some of the language and violence for the DVD release to avoid complaints about the PG rating.