Other mistake: At the end of the episode after delivering his complex greeting to the Jarada, Picard and Riker head for the Turbo lift. Just as they are about to enter, Picard opens his mouth really wide as if making a funny face to someone inside the turbolift or maybe quietly celebrating a 'successful take' given the bizarre nature of the greeting! Although it was filmed from the side it is quite easy to spot.
Visible crew/equipment: Near the start, just as Riker says "You have all the time you need.", the engine consultant stops and turns around. As he does so, look above his head and you can see a microphone hanging down. The camera quickly adjusts so it's not visible any more. [Not visible in the remastered Blu-ray version].
Continuity mistake: The lipstick on Picard changes between when he is kissed, and later when he leaves the holodeck (it's darker and on and above his lips.) Also, and more importantly, being holodeck matter, it should have disappeared after re-emerging in the real world.
Continuity mistake: When Picard tells Q in the final scene "Get off my ship." Q is standing to his right, and Picard has his head turned towards Q. But when the angle changes to show Q's response, Picard is standing facing Q directly, and without his head turned.
Continuity mistake: When Data brings up a picture of the original Kirk-era Enterprise in conjunction with the log talking about the water-intoxicant, the picture is of the refitted movie-style Enterprise - not the original round-nacelled model that was involved in the events of "The Naked Time."