Plot hole: Dr. Adam Soong is initially presented as a discredited scientist, banned from the scientific community; he gets debarred and his funding revoked. And it's not an internal matter; he is publicly exposed for it. His daughter in episode 6 even finds out this information on Google. Several news articles call him "mad scientist" and such. However, this same person at the same time throughout the rest of the season has every bit of pull and influence, not just through undercover channels, but is treated with the utmost honor and deference by the NASA PR people at public events.
Other mistake: The show is supposed to take place in the same timeline as Next Generation, but the presence of a young Guinan in 2024 who has never met Picard before contradicts the TNG Season 5 finale, Time's Arrow, where Picard met Guinan (looking like Whoopi Goldberg, who also appears here) in the XIX century. When asked about this contradiction, show writer Terry Matalas said that the Federation actually never happened since the 'future' was changed by Q's actions, so Picard never traveled back in TNG meeting her. However, that is never referenced in the actual episode or the season finale, and would create other paradoxes, especially since in the very same episode there is a reference to Star Trek IV. The same punk from back then rubs nervously his neck remembering Spock's nerve pinch; the TOS cast would have not visited XX century Earth in the way shown in the movie if the Federation never came into existence.
Plot hole: In this episode, the 'Watcher' displays the power to possess/mind control people at will, jumping from body to body to set up the meeting with Picard. Forgetting the fact that this usage of power for such a menial task is actually detrimental to what she wants to do (it leaves more evidence, by unnecessarily messing up with minds just to tell the guy to go from A to B), this would have been super-useful for the rest of the season, but she just never ever uses it again, not even a nerfed version of it.
Factual error: Picard tells his manservant that Laris and him are more needed at the Chateau than with him in space because there's less than a month left before the harvest. Without being privy to the knowledge of a XXIV century agronomist, grapes at the Chateau are grown with traditional methods in a climate that looks reasonably similar to our own. From the establishing shot at the beginning of this episode and what we could see by his walk in the vineyard in the previous, there are no grapes forming, ripening, not even flowers. Hardly the look of a vineyeard close to harvest. (00:07:00 - 00:36:10)
Continuity mistake: When Lieutenant Narissa Rizzo appears in holo form in the last scene of the episode, the two pins of her rank on her right breast initially are attached together (as if the two magnets have been placed too close together and ended up being pulled to one another). During the first part of the scene they go back and forth between being joined and separated. (00:40:45)
Continuity mistake: When Picard walks out of the archive building he's shocked to see Dahj, who is wearing boots with a chunky heel (also seen when she followed Laris to a bedroom for "a good night's sleep"). After their talk Dahj and Picard head to the roof, and while Dahj fights the Romulan assassins she is wearing flat soled boots, but in at least one shot as she kicks an assassin her chunky heeled boots reappear. (00:28:15 - 00:32:45)
Continuity mistake: Right at the beginning, the camera zooms into the Enterprise through a view that shows 3 windows of equal size. In the rest of the scene, Jean-Luc Picard and Data are playing poker in front of two side windows and a central window that is wider than the other two combined. (00:00:30)
Character mistake: Dr. Agnes Jurati, one of the Federation leading scientists, is bored during the hyperspace travel, and so she chats a bit with the captain. While she gives her quirky speech, she casually mentions that "there are over 3 billion stars in our galaxy." She's not wrong, technically, but the number of stars in our galaxy is estimated between 100 and 400 billion. She is way off. (00:08:40)
Character mistake: Picard and Data are playing poker at the beginning of the episode. Before the actual faces of the players are revealed, we are shown close-ups of their hands as they put down the chips for the bet. When Picard says his first line in the series it's "See", and it is literally true, because in the previous shot, Data put his 5 cards down on the table and face up. (00:00:50)
Plot hole: Some part of the dialogue between Jean-Luc Picard and Admiral Kirsten Clancy must have been left on the editing room floor, because when Commodore Oh refers to the conversation the two had, says that "He referred to Zhat Vash by name." He did not. (00:21:30 - 00:37:30)
Continuity mistake: At Raffi's home, when she rips into Picard and tells him how she had lost her security clearance and more, Raffi stands up and grabs both the bottle and red glass then walks away from the table. In the overhead long shot, as Raffi walks down the steps the bottle is in her right hand, and her left hand is empty, then when Picard stands up we see Raffi's red glass is back on the table near her chair. (00:10:30)
Continuity mistake: Picard, panting after the walk through Vasquez Rocks, tells Raffi he's sorry. Michelle Hurd in the first close-up and wide shot has her hand on her knee, with the arm coming up from the inside leg. In the close-up that follows, she is leaning her face on her hand as it comes up from the other side of the leg. From then on her head position changes inconsistently as well. (00:14:05)
Continuity mistake: Dahj kills the 3 dudes who tried to subdue her putting a bag on her head. When the third one kicks the bucket, notice how he falls with his leg spread, a tablet-like object rolls on the floor touching the carpet, and on the carpet itself there are several pieces of debris and shattered glass. In a following shot the bag that was totally not hiding a stuntwoman doing agile flips and kicks is removed from her head and we see Isa Briones standing barefoot. She stands on that same carpet but it has been sanitized from anything sharp, while the aforementioned Romulan lies on the floor positioned differently and the 'tablet' is gone. (00:06:00)
Continuity mistake: Picard during his conversation with Raffi on the bench outside the Federation HQ building is at one point sitting with his arm folded, no tablet in hand or lap. By the end of the convo, without him moving, the tablet reappears in his arms, clutched to his chest. (00:04:55)