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 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.
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 asked by Capt. Picard if Data has his emotion chip equipped, Geordi La Forge responds that "He didn't take it with him." Star Trek: Generations had established that Data's emotion chip had been fused into his neural net and could not be removed by any means. Of course, it is quite possible that a way to remove the chip had been discovered between the two movies, but it is never addressed or explained by any means.
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.
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.
In the first scene on the Bridge of the Enterprise, Riker's beard is obviously not his real facial hair (compared to his real beard in other scenes). It is larger, there are no signs of gray and the mustache extends several inches off the sides of his lips. A possible explanation is the scene was filmed later in production after Jonathan Frakes shaved off his real beard.
The affect of the planet's rings caused Geordi LaForge's eyes to grow back. In the season 2 episode "Samaritan Snare" it is established that Capt. Picard has an artificial heart. The metaphasic radiation would have caused him to re-grow his own heart.
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 tells Anij that what they Baku have would be worth more to some than gold pressed latinum. But nothing would have a value in a society without money unless you needed to barter for it, in which case the value would be based on what the seller needs and is willing to give for it.
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 items below may give away important plot points.
Before Ru'afo gives the order to deploy the collector, he gives the order to "separate the Starfleet personnel and put them in the aft cargo bay. See that Picard joins them." While the So'na were using their ship to host other Starfleet personnel in the beginning of the movie, it is established that these personnel were transferred to the Enterprise, except for Admiral Dougherty (who by this point was dead). Since none of the other members of the away team were tagged on the surface of the planet, Picard should have been the only Starfleet officer on the So'na ship.
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.