Revealing mistake: On the second day aboard the Orca, Quint fires a barrel into the shark and they start to pursue it. There are some overhead shots of Quint on the pulpit and during some of these shots, you can see some sort of tow cable. It's visible on the bottom of the screen between some of the supports of the pulpit.
Other mistake: When Cooper is first examining the remains of Chrissy, the medical examiner takes her remains out of a cupboard of sorts, and carries them to a table. He doesn't struggle, and lifts the tray rather easily yet when Hooper is describing the remains of the body, he says "The torso has been severed mid thorax, the right arm has been severed above the elbow, the left arm, head shoulders sternum and portions of the rib cage are intact" The mistake is that Hooper describes way more of the remains than can have fit in the small tray and or be lifted so easily by the medical examiner.
Continuity mistake: When Quint is up in the crow's nest, Brody tells him "Let Hooper take a turn" (regarding chumming the water). When Quint, while looking down at Brody, says "Hooper drives the boat, Chief", his right arm is across his body and looks to be resting on the safety railing of the crow's nest. When Brody looks up at Quint, his right arm is away from his body and is just holding onto the crow's nest railing with his hand.
Character mistake: Chief Brody and Hooper go to the wharf to dissect a large tiger shark and examine the contents of its stomach. Finding nothing unusual, Hooper recommends they go offshore that night to search for the real killer shark because "he's a night feeder." Coming from a marine biologist, that remark really makes no sense. Hooper knew that, in addition to eating Chrissie the midnight swimmer, the shark also ate Pippin (the black Labrador retriever) and the Kintner boy in the middle of the day at a public beach. Based on all available evidence, the shark was no more likely to feed at night than in broad daylight.
Revealing mistake: When the shark swims away with three barrels, in the next shot all three men are standing at the back of the boat watching it. Look at the sea in front of them and you can see one of the barrels waiting to pop up.
Continuity mistake: As Brody flips through his shark book, he pauses to look at a photo of two fishermen and their catch. The photo is on the right page, but in the previous shot where we see Brody stop flipping through, there was no picture on the right page as seen in a reflection on his glasses, just words. (00:25:30)
Continuity mistake: Throughout the movie we see a number of underwater scenes with the camera shot looking up under the swimmers where the water is very clear and blue given how many swimmers we can see from a distance. The only problem is that the water everyone is swimming in is greenish and cloudy as can be seen with the camera shots that are half in and out of the water right along side of the swimmers.