Other mistake: When Dean compliments Les on the great driving making up for a year of boredom, Les looks around up in the air in a direction away from the car, then suddenly says "there's a scratch on my car." He then looks down on the car at the scratch and shows Dean. He said the "scratch" line before actually seeing it.
Revealing mistake: When Les' mom is doing the laundry and "finds" the Test Failed notice, you can see that she actually had it in her hands the entire time. She transfers it from one hand into the other, then puts the hand with the paper into the pocket and pulls it right back out, then knocks it off onto the floor, and acts surprised.
Continuity mistake: In most of the film, the Cadillac has personalized license plates reading "GRANPA" or "GRANDPA", except at the drive-in restaurant, where the license plates read "BDR 529" (also the number on the Bluesmobile in "The Blues Brothers"). When they drive away from the restaurant, the "GRANPA" plate seems to shimmer and move around - an apparent traveling matte made in an attempt to correct the error.
Continuity mistake: When Les pushes the Cadillac out of the garage, he leaves the driver's door open as he goes to turn off the garage light. When the car continues to roll across the lawn and through a hedge, the door is closed and he cannot open it in time to stop the car.
Factual error: There is no way the DMV will pass one twin based on the fact that his twin passed, even though they ended up taking his license. The mistake was giving it to him in the first place based on the fact his twin passed.
Plot hole: Les comes home and tells his parents he passed his test. A scene later, his mom is going through the laundry and takes the test out of his jeans pocket. How did it get there when Les is still wearing the same outfit he wore to the DMV?
Factual error: Les' friends and sister are shown getting their real licenses (the actual cards, with photo) right away, the same day as their test. But in California, where they live, you don't get your license for six weeks or so - until then, you get a paper printout.
Audio problem: When the drunk driver steals Les's Caddy his nerdy friend says you can't take this car and Les says "He took our car, we're gonna take his" but his mouth doesn't move at all.
Factual error: When Les is pulled over by the cop, he has Les open the trunk to reveal Mercedes in there passed out drunk. Les told the cop she was his date. The cops then get a call to report to the scene of the riots. The cop told Les he got off lucky. No cop will spot a body in the trunk of the car and let the driver off scott free. Even if the person in the trunk appeared drunk, Les would have been put in handcuffs and taken to the station until it was all cleared up. Even if they were called to handle the riots, the cops would have told dispatch what was going on and taken care of the Les problem instead.