While at the lake, Data says that he was designed as a flotation device. This is not true. In Star Trek: The Next Generation: Descent: Part 2 Geordi reminisces about when he and Data went sailing on Devala Lake. Data decided to go swimming, jumped out of the boat and sank straight to the bottom. Because of his lack of buoyancy Data was forced to walk 1.46 kilometers along the bottom of the lake to get back to the surface. Even if he has altered his body to accomplish this task, he was not "designed" that way by Soong.
When Anij slows down time (the water falls very slowly, the bird's wings flap slowly) she picks a flower and then blows the spores off of it. She's slowed down time, but when she picks the flower the other flowers next to the one she picks move back and forth at full speed.
During the phaser fight in the holoship, A Son'a soldier is shot and rolls off the roof of the hut, and when Picard ends the simulation, the soldier is not lying on the floor stunned when we see the full holodeck revealed, unless he was a hologram too.
When Data opens the door to the cloaked holoship, he and Picard and Anij are kneeling on a raft -note the paddle in Picard's hand. To enter the holoship they need take only 3 steps up (about 3 or 4 feet). But about one minute later, when they come under fire and Picard pushes Anij back out the door through which they came, the surface of the lake is at least 15 feet below the doorway.
When he discovers the holoship hidden in the lake, Data opens a dam and lowers the water level. But when he boards the raft with Picard and Anij, the pier is still at the correct height for the normal water level.
After Picard rescues Anij from the water, Data says, "In the event of an emergency, my body can be used as a flotation device." Data then "rises" above the water as if he is inflating. The platform that raises Brent Spiner higher in the water is visible through the water underneath him.
When the crew meets Picard in the Captain's Yacht, the captain tells Geordi and Riker to take the ship and transmit a message to the Federation. If you look at Geordi's collar you see a gold and two black pips (rank insignia). He should have two gold and one black pip for his rank of Lt. Commander (one gold and two black pips does not belong to any rank in Star Trek).
Picard sings "heart" instead of "foot" in the second chorus of "A British Tar", despite having the lyrics displayed in front of him on his computer. Considering the stress he's under, he does an excellent job.
Data used a contraction in the scene where the cave collapses, saying "it's safer over there." Although the series established in later seasons that Data can not use contractions, he had the emotion chip installed into his positronic net in Star Trek: Generations which would likely allow him to not only experience emotions, but to use contractions as well. Episodes in early seasons of Star Trek: The Next Generation showed Data using contractions. Also worth mentioning that although Geordi tells Capt. Picard that he didn't take his emotion chip with him, it is possible he reinstalled it during the course of the movie.
Admiral Dougherty tells Picard - admittedly, perhaps deliberately - it could take ten years for the Son'a to start feeling the effects from the metaphasic radiation; yet the entire Enterprise crew, newly introduced to them, begin to feel effects almost immediately upon entering the planetary system (i.e. Geordi's eyes). Picard doesn't contradict this statement by noting the example of his crew's reaction to the radiation in arguing more strenuously against the Son'a plan, which would make unnecessary the subsequent actions of him and the Enterprise crew.
When Picard is first meeting with the Baku, there are children playing a hacky-sack type game in the background. During two short shots, the children can be seen making kicking motions but there are no hacky-sacks to be seen.
The goof item below may give away important plot points.
Incorrectly regarded as goofs
During Data and Picard's phaser fight inside the holoship, the disruptor shots from the Son'a weapon do not disrupt the holograms as we see the Son'a Commander do later in the movie to reveal that was fooled. In fact, they chip walls and splash water. However, the holoship was designed to be a 'floating holodeck' to transport unwilling colonists, so the simulation would likely be programmed to react in a realistic manner to weapons fire so that the deception isn't accidentally discovered; likewise, the crew of the Enterprise may not have had time to reactivate the 'reality mode' when they changed the simulation.