When Dr. Crusher is explaining Shinzon's genetic engineering, she speaks of RNA and DNA in the appropriate instances. However, when she mentions that Picard is the only one with compatible DNA, her lips say "RNA".
During his confrontation with the hologram of Shinzon, Captain Picard states "Your heart, your hands, your eyes are the same as mine." While Shinzon is a clone of Picard and most of this is true because they share the same DNA, Picard actually has an artificial heart. It was established on Star Trek The Next Generation that he lost his biological heart as a cadet when he was stabbed through the heard by a Nausicaan.
When speaking to the wedding attendees, Data addresses them as ladies, gentlemen and transgendered species. The proper term for a species with no gender is androgynous. Transgendered is going from one gender to another.
During the wedding reception Data refers to jazz as an "archaic musical form", even though everyone in the room seems to be familiar with jazz and is dancing correctly. Even Worf knows who the composer of Blue Skies is. Jazz has been featured prominently throughout TNG's seven seasons, as well as in DS9 and Voyager.
Some "facts" about Picard established in episodes of the TV series Star Trek: The Next Generation are overlooked or ignored (e.g. in the series, Picard had hair at Starfleet Academy and no longer has his own heart, it having been replaced after an incident there). There are other elements of discontinuity from the Star Trek series (Worf's transfer to the Diplomatic Corps and Wesley Crusher's resignation from Star Fleet are both ignored and his departure with the "Traveler" during the seventh season), although some of these are explained away in deleted scenes.
At the end of Picard's first meeting with Praetor Shinzon, Shinzon orders the lights to dim back to their original setting; however, the lights begin to dim when he says, "Computer," and before he gives the order for the lights to return to their normal setting.
Towards the end when the Scimitar deploys the targeting arms they open to about one third then, just before Picard says "How long until he can fire" the shot shows the targeting arms completely collapsed.
Moments before the Enterprise rams into the Scimitar Geordi reports that the Scimitar's shields are still at 60%. If this were the case the Enterprise wouldn't be able to get that close to the Scimitar without severely damaging the ship, which of course doesn't happen until the actual collision.
When the Enterprise first arrives at Romulus, Picard rises from his chair to walk towards the viewscreen. In the wide shot, the camera pans slightly too far right, revealing the leg and profile of a crew member in street clothes and shoes standing on the bridge.
During Data's briefing, we see Remus as being in an outer orbit compared to Romulus. But in a later surface shot of the Reman mines on the dark side of the planet, a sunlit Romulus appears in the night sky, implying that Remus is in the inner orbit.
The observation lounge doors do not line up with the doors that feed into the observation lounge on the bridge. The placement of the doors would send someone right into a wall behind which would be a turbolift car (however in the film the doors lead to a corridor that wouldn't be there).
During the Briefing en route to Romulus, Data advises the Senior Staff about the planet Remus, and he describes Remus as being in a tidally-locked orbit around its sun (meaning one side of the planet always faces its sun) and because of the extreme temperatures on the day side, the Remans live on the dark side of the planet. In a planet that is tidally-locked to its sun like Remus, the dark side would technically also have extreme temperatures as well, on the cold side. The most ideal habitable zones on a planet like Remus would not be the dark side of the planet, but actually the terminator sides (where it transitions from day to night) as temperatures on those sides of the planet would be temperate and most optimal for most forms of life.
The orbits shown of Romulus and Remus as shown in Data's briefing are not possible according to Kepler's Laws of Planetary Motion. The gravitational pull of two planets passing in such close proximity would have long ago dislodged one of both from its orbital path.
After the collision between the Enterprise and the Scimitar, Shinzon orders his ship to back up, as if they were two cars crashed in a parking lot. The Enterprise remains stationary violating all laws of physics as the Scimitar backs away from the combined wreckage.
Although Shinzon and Picard have identical DNA, speech patterns are not genetic. It is a learned behavior and a product of the environment one grows up in. Although he was probably taught by the Romulan government to speak with a British accent from birth, he more than likely would have become a product of his Reman surroundings and "grown out" of the accent and begun speaking with a similar accent as the Remans or Romulans.
During the wedding scene, Picard says it has been the privilege of Captains to conduct marriages "since the time of wooden ships." This is a myth. Captains were never accorded this right and cannot do so today, unless otherwise ordained or licensed.
Just as the Scimitar is about to fire upon the Enterprise, Captain Picard calls for the shields to be raised even though the ship has been already at "Battle Stations" status for some time. However, "Battle Stations" means that the crew members are at their stations; it does not mean the shields are up.
Lt. Commander Worf, having been promoted to this rank in Star Trek: Generations, should be wearing the correct rank pips on his collar. However, he is shown throughout the film only wearing 2 pips, his former rank of Lieutenant. Lt Cmdr Worf is, in fact, wearing the three pips signifying his rank as Lieutenant Commander. The third pip is difficult to notice because of his goatee.
The movie takes place after the end of Star Trek: Deep Space Nine where Worf is given the position of Federation Ambassador to Qo'nos. Yet he appears here in a standard uniform and with no mention of his high ranking position.
Data states that the inhabitants of Kolarus III are a "pre-warp" society, but yet the Enterprise has no problem with "contaminating" it with its presence in direct contravention of Starfleet's Prime Directive.
In the first few shots of the Argo buggy, as it leaps from the back of the craft and makes several sharp maneuvers, it is apparent from its stiffness that the figure in the rear seat (supposedly Worf) is actually a dummy.
The goof items below may give away important plot points.
After Data expels himself out of the destroyed forward sections of the Enterprise, the film soon cuts to shots of Data outside the Scimitar and then the Scimitar bridge. However, in both of these shots the forward sections of the Enterprise are seen to be intact.
The Remans board the Enterprise on the lowest deck where the shields are down. Yet several scenes later the Viceroy falls the height of several decks despite supposedly being on the most bottom deck. However, the Remans were likely to ascend several decks in their search for Picard. Some of the signs on the surrounding doors even show they are on a deck higher up then the lowest one.
Data's emergency transporter device can only be a transporter beacon of some type since it lacks the essential hardware to be able to transport someone on its own. So, it would require a functional transporter to tap into to operate properly. Yet, in the climax, the Enterprise transporters are fused (but it somehow operates anyway).
When Shinzon makes his final lunge at Picard, the piece of ship Picard uses to impale Shinzon can be seen to shift laterally across Shinzon's chest, an impossibility if it had impaled him through the chest.