Near the end, Data tells Riker that the Enterprise would have to leave for the Neutral Zone in 48 minutes, in order to reach it in time at maximum warp (which, for the Enterprise, is ordinarily about warp 9). A little later, that deadline has gone down to 17 minutes. Yet when Picard thereafter retakes command, he orders the ship to proceed to the Neutral Zone at warp 6, in which case they would not reach it in time.
While Geordi and Dr. Crusher have a conversation over the intercom, a light haired female extra playing a medical officer can be seen in both the sick-bay and on the bridge at the same time.
When Troi, Yar, Data and Riker beam down to the planet for the first time, Riker clearly is carrying an item. When they meet the female leader of the planet, they walk in and Riker is not carrying the item. When they are escorted to their accommodations, Riker is seen carrying the item again.
When Yar, Data and Troi beam over to meet with the Odin survivors for the first time, both Yar and Troi begin the trip with their arms behind their backs. When they rematerialize, however, they have their arms at the side. Also, the positions and stances of all three change slightly during transport.
When Mistress Beata is talking with Riker, in one scene, her right arm is at her side. An instant later, her right hand is massaging the back of his neck.
Riker says that it is not Starfleet's policy to interfere with the domestic affairs of other cultures but notes that they can interact. Interaction with a system by its very nature alters that system, no matter if it's intentional, no matter how small the impact may seem.
Captain Picard is walking past the Holodeck when he gets hit by the snowball. The Holodeck can create "real" (i.e., non-holographic) objects depending on the programming parameters, and in this adventure creates an actual biological virus.
It is never explained how or why the Federation has any sort of relationship with Angel One. The original Prime Directive forbade interference with the development of pre-warp civilizations and Angel One is said to be at a roughly mid-20th century Earth level of technological development. However, for ST:TNG, the Prime Directive was re-written (see Trivia) to prohibit interference with any culture's development.
Right before returning to the Enterprise, Data contacts the ship with the words "Standing by to beam up", on which he is beamed straight away. It is sometimes thought that he is telling the transporter operator to stand by. However, that is incorrect. He is saying that he is standing by, meaning he is awaiting transport as soon as the transporter operator is ready.
Data says the inhabitants of Angel One are similar in development to mid-20th-century Earth. But they have a disintegrator beam, indicating a technology far in advance of the 20th century. Note that he said "similar," not "exactly the same as."
Mistress Beata refers to her planet as Angel One. This is very odd, as if someone from Earth would refer to their planet as Sol Three.
While it might seem odd to an Earthling, it is a logical name and probably sounds completely normal to an alien living on Angel One.
In the last few seconds, before the closing credits, the main bridge turbolift is seen in a locked-open position, yet no one is passing through the doors.
After the away team leaves their room, Trent sprays himself with a perfume bottle. The bottle has a rubber bladder pump mechanism, yet the sound heard is of a continuous aerosol spray.
When Riker and Beata are alone together in her quarters, Beata places her glass down without looking and a hand can be seen taking the glass from her.
Even with the Holodeck's reputation for malfunctioning, it is difficult to accept that anyone would program an infectious biological disease into a recreational program. (The virus is a holdover from the script's first draft, where Wesley and co. went skiing on a real planet.)
Data is unfamiliar with the concepts of perfume and aphrodisiacs. It has been established that at this point Data has been active for 26 years and has both an encyclopedic knowledge and a fascination with humans. It is inconceivable that he would not know those terms.
Trent is given the duty of carrying out the "revolutionaries'" execution. If Angel One is, as Data stated, similar to the old gender dynamics of Earth, simply with the genders reversed, a man on Angel One would not be in charge of carrying out punishments, in the same way that women would not have been put in charge of carrying out capital sentences in eras past.
Beata states that, because the Enterprise crew refuse to take the Odin survivors with them, she must sentence the survivors all to death. Given that the people of Angel One never expected a Federation starship to come retrieve the survivors, they should have already received whatever sentence Beata deemed fit.
When Dr. Crusher is working on Picard in his quarters, she mentions the scent coming off him. He explains that he also noticed that same scent after he and Worf were hit with the snowball. All of a sudden, Crusher knows *exactly* what to look for. Wait - what? How did THAT identify the virus that she had been looking for over several days without success?
When Riker meets with Beata and Ariel, the shadow of a boom mic moving around can be seen on the wall behind them.
Data states that the Odin was "not a starship" and thus its crew is not bound by the Prime Directive. A starship, by definition, is a vessel designed for travel between star systems, which the Odin very much was. What Data meant was that it was not a Starfleet ship.
During his impassioned plea to Beata, Riker states that no force in the universe can hope to stop the force of evolution. That is incorrect. There are many ways to stop evolution. The Enterprise encounters such things regularly.
Data asks Riker what they are to do if the inhabitants of Angel One deny the existence of the survivors from the Odin, to which Riker says, "Let's not look for problems." Asking for a planned course of action in the event of a very possible obstacle is not "looking for problems."
When the Enterprise begins to search for survivors of the freighter Odin, Picard orders La Forge to break fixed orbit. But a fixed orbit means that the ship remains above a given location constantly. The shot of the Enterprise shows the planet is rotating in one direction, while the Enterprise is moving in the opposite direction. That is a counter-rotational orbit.
There is not one paper tissue on the entire Enterprise for all this sneezing (even Klingons).