(at 27 minutes, 16 seconds) As Troi is preparing to counsel the young 20th century lady, we see a model of the Enterprise-A from the movies, painted flat gray. However, the warp nacelles are put on the struts backwards, with the pointy end facing fore, and the blunt leading edge facing aft.
Ralph Offenhouse at one point states "A military career has never been considered to be 'upwardly mobile'". While perfectly in character, it is factually incorrect. A great many historical figures rose from humble backgrounds to positions of the great power, wealth and influence through the military, like Napoleon, the Duke of Wellington and Dwight Eisenhower, to name but a few.
Troi explains to Claire that the image she sees is her "great-great-great-great-great grandson." With the passage of more than 350 years, far more than 7 generations of her progeny would have been born.
When Data tries to see into the frozen cryogenic chamber, it appears he uses the heat of body temperature to remove the frost to see the contents, which would not be possible unless the internal temperatures regulated by the thermal control system referred to in Birthright, Part I (1993) extend to his extremities.
However, it is entirely possible that (similar to using a scraper on a windscreen) simple pressure would remove the frost, and Data is quite capable of exerting sufficient pressure.
When Picard walks the floor in sickbay, the floor can be heard creaking. The floors in the Enterprise are not wood and should not creak when walked on.
When Captain Picard orders Data to sickbay, Data looks to Commander Riker, who gives an approving nod before Data leaves the bridge. Data would not require approval from Riker, as Picard outranks him and is the commander of the vessel.
When researching the fate of Clare's children back in the 21st century, Troi tells her "There must be a record somewhere." While the reassurance may be in character, in the Star Trek universe, there was a nuclear war on Earth after Clare was frozen, which must have destroyed plenty of family records.
Picard is seemingly annoyed with Data's decision to rescue the three frozen humans from the derelict orbiter, and is more so with Beverly's decision to thaw them out. This is rather immoral and downright unprofessional of the captain as part of his duties would be to save lives. Here, Picard seems to prefer discarding those lives all because they got in the way of a critical and intense situation.
When Data discusses the cryogenic survivors past occupations, he states that Clare's occupation as "homemaker", then states she must have built houses. With Data's vast knowledge he should have known that "homemaker" is not someone who builds houses.
When Deanna first visits Clare, the door chimes (the usual sound when someone is requesting entry), then the door opens without a response. If she was not going to wait for a response to enter (a response which may never have come, as Clare likely would not be aware of the chime's significance), Deanna would not have rung it.
Picard, Riker, and Data are shown meeting with two of the revived 20th century humans in the briefing room. They are then shown returning to the bridge via the port turbolift at the rear of the bridge. The briefing room is located aft of the bridge on deck one (typically accessed via the doors at the starboard rear side of the bridge). They would not have needed to take a turbolift to reach the bridge from there.
Riker orders security to get Offenhouse off the bridge. Not only did they not follow their orders, but they let go of him and allowed him to interact with the others. The bridge officers, including the Captain, said nothing until much later.
It seems unlikely that an unauthorized person:
1. could take the turbo-lift to the bridge
2. could get onto the bridge without anyone (officers, captain, or security) noticing
3. would be allowed to remain for even a minute or speak/interfere with it
4. would not be removed immediately by security.
If the cryogenic station was a 20th Century orbiter, there is no way it could have gotten into deep space. Also, none of the lights would still be working, considering they had been on for three centuries. In the outside shots of the station, the "solar panels" are shown to be clearly damaged/broken, hence it would be unlikely that they would still be operating after three centuries away from suns to provide them energy.
Data is seen pressing a number of commands on the replicator before telling Sonny to speak his food order. The replicators never require these manual commands before or after. Rather, the person simply approaches the replicator and speaks.
The Enterprise's sensors, as well as Data's tricorder, should be more than capable of identifying biological matter aboard the derelict capsule, yet Data and Worf are surprised to find the frozen humans aboard.
Shadows on the wall behind Picard, a few seconds after Riker orders shields up.
Riker shows little interest in the ancient Earth ship they encounter, which is a somewhat peculiar attitude considering he and the rest of the Enterprise crew are usually fascinated by all manner of things from Earth's past.
As Data and Worf beam onto the ancient capsule, Data checks his scanner and announces there to be "minimal oxygen atmosphere". It would be a little too late for Worf had it turned out the atmosphere wasn't breathable.
While Riker is talking to the three humans from the cryo ship, Picard pages him. Instead of tapping his com badge to answer, Riker walks over to a panel to do this. For the entire first season, Riker has always been seen tapping his badge to summon someone or answer, never walking over to a panel like this. However, the writers made this part of the plot so that Offenhouse could see Riker do this, then later on bother Picard.
The Enterprise crew is presented here as unusually incompetent when dealing with older technology and culture. Data is somehow unfamiliar with the term "homemaker" (despite having access to the Federation's historical records, which would undoubtedly contain a definition of the term even if it had fallen completely out of use by the 24th century) and Worf is unable to operate a door simply because it does not open automatically when he approaches.
Picard asks the Enterprise computer how long it will take to reach the Neutral Zone. Data is right there and can calculate faster than the computer, and in any event, it's a pretty basic question that even the helmsman or most bridge officers could answer. There seems to be no reason for Picard to address the question to the ship's computer.