Continuity mistake: As Stripe bites at Pete, Pete jumps back. A second later, he jumps back again.
Continuity mistake: When Gizmo goes to Rand for protection, he's upright as Rand's hand comes around to comfort him. A second later, he's in a reclined position, with Rand scratching his stomach.
Continuity mistake: As Billy opens the box, Gizmo pops up and goes back down. When the angle changes, Gizmo is starting to go down again.
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.
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.