Character mistake: When Will Riker comes to Troi's office and kisses her, Troi says that she has never kissed him with a beard before, but in the third season episode "Menage a Troi", Troi does in fact kiss a bearded Will Riker, while on leave at Betazed, in the Fifth season "The Outcast", and again in season six's "Man of The People."
Continuity mistake: When Capt. Picard, Data, and Anij are going to the holoship, they start off from the landing dock with (facing them) Data on the left, Capt. Picard on the right and Anij sitting. This is all well and good but after Data opens the door to the holoship, they magically switch places.
Continuity mistake: Picard has a cut on the right side of his head in the scene where he, Worf and Gallatin are on the Son'a bridge. It is most visible when he says to Gallatin "Take us into position". After Picard takes the rifle from Worf and Gallatin says "We're approaching the collector", the cut on Picard's head is gone. After he has beamed over to the collector, the cut is back and is bleeding a lot more than it did before.
Other mistake: When Data discovers the holoship in the lake he reduces the amount of water in the lake significantly (The ship is now clearly visible, standing out several meters). But even so the banks and the landing stage are still in their correct position. If the surface level had been reduced the landing stage should have been standing out of the water, too.
Continuity mistake: When Picard and his team beam down to the planet for the first time to "rescue" starfleet officers you see a couple of kids playing a ballgame. These balls were CG, however, in a next shot when Picard is talking to the Ba'ku people you see the kids still playing the game in the background, but without the CG balls.
Revealing mistake: The scene on the bridge where Picard tells Riker and Troi they have two days to become experts on the Son'a was filmed towards the end of shooting. You can tell this because Riker's beard and moustache are obviously fake. Maybe that's why he doesn't have any lines - too worried about his moustache falling off.