When the Enterprise is travelling at warp, we can see through a window in the Captain's ready room no stars streaking by that we normally get during the warp effect. It can also be seen when Commander Riker is about to leave and the viewscreen is visible on the Bridge.
There is a discernible difference in the blue of the scientific division uniforms, most notably between Doctor Crusher and Lieutenant Commander Nella Daren's uniforms. The darker blue, or teal blue, will carry over to the new uniform redesign. There is no explanation to the difference in colors.
When Picard and Daren are in the Jeffries tube, she rolls her keyboard onto the floor, then it is seen on stands.
While the above is true, Lt. Cmd Daren can be heard assembling the stand while Picard is on screen.
Picard's hands are clearly being doubled, when he and Nella improvise with Frere Jacques. (His hands are a lot lower and straighter than they ought to be, if he was really playing the flute.)
Some obvious matte work to simulate a long access tunnel when Picard and Daren play music in the Jeffries tube.
The second time Picard visits Nella Daren in her lab, a crewman enters through the doors behind them. The doors don't close all the way; they jam and leave about a 15 cm (6") gap between them.
As the camera zooms out while Picard and Daren are playing, Daren's portable piano can clearly be seen on a stand.
In the opening scene, when Picard enters the bridge from the aft turbolift, the door decal is the wrong colour. It is light purple, which is used on crew quarter/rec decks. It should be the maroon typically used in non crew quarter / recreation areas of the ship. This was noticed in the remastered HD version.
The roll up keyboard has too many white keys/too few black (gold) keys and therefore does not reflect an actual piano/keyboard layout and would not be playable.
The Enterprise beams the rescue team into a hot, dusty firestorm without basic safety equipment like work gloves or goggles, let alone respirators, self-contained air supplies, or sealed, heat-resistant environment suits.
Knowing that they would be short on time for an evacuation, why did they not employ shuttlecraft in addition to transporters? This would have given them a much larger margin of safety.
Lt. Commander Daren is in charge of an entire department on the Enterprise. It makes no sense that Captain has not met someone in such an important role.
Picard discovers that all power to the replicators was shut down so that it could be allocated to stellar cartography, which was done by Nella Daren. It is unbelievable that Daren, a lieutenant commander, would have been able to accomplish this feat without first obtaining Picard's permission.
When Picard is in his quarters drinking his tea, before Lt. Cmdr. Nella Daren enters the room.
When the away team is about to beam down, Commander Riker is heard (off screen) saying: "ensign, keep a continuous lock on all away team members"; however, when the officer is shown, she wears the rank of Lieutenant. The commander would never make that type of mistake.
When Picard comments on Lt Cmdr Daren's use of the "second arpeggio" in Frédéric Chopin's Piano Trio (op.8), he notes that she replaced a minor triad with a fully diminished triad. This is a recompositional move no professional musician would ever make.
Data says to the captain that there will be a concert of Frédéric Chopin's piano-trio in G. However, this is not played in G minor, but in A minor.
When Captain Picard first meets Lt Cmdr Daren she is wearing 3 full pips on her collar. That is the rank insignia of a full Commander. She should only have 2 full and one hollow pip on her collar.