Character mistake: Neelix says that the Great Wall of China, prior to the 22nd century, was the only man-made object visible to the naked eye from orbit. This statement has been long-since debunked, but it's not unreasonable to think the myth perpetuated itself in the future. Still wrong though.
Character mistake: Captain Janeway states the temporal differential ratio is .4744 seconds per minute (meaning for every minute on the surface, only .4744 seconds pass on Voyager). Tuvok calculates that 30 minutes on Voyager would be 2 days 11 hours and 47 seconds. However, at that ratio, 30 minutes would be 2 days 15 hours 14 minutes and 15.9 seconds. Later Tom (who was on the surface) says the last two months for them was only 2 days for B'Elanna (who was on Voyager). But with the temporal differential, 2 months on the surface would be less than 12 hours on Voyager. 2 days on Voyager would be almost 253 days (almost 8.5 months) on the surface.
Character mistake: At the start, Captain Janeway reveals that she has discovered Tuvok's birthday. Then she says it's not long before he hits the big 3-digits (meaning 100 years old). However, Tuvok was 107 when he infiltrated the Maquis and they've already spent 6 years on Voyager. Even if she didn't discover his birth year and was only guessing. Tuvok would not have replied with "indeed" but pointed out she is wrong.
Character mistake: When the now human Borg children are sculpting, one of the children tell Seven he made a cube that's precisely 1/1,000 the size of a Borg vessel. Borg Cubes are described as 3 kilometers in length, if not bigger. 1/1,000 of that would be more than 9.8 feet in length. 1/10,000 the size would have been a more accurate description.
Character mistake: Species 8472 was trapped on deck 11, behind a force field. When the ship comes under attack by Hirogen, Seven is instructed to get down to deck 11 with a supply of nanoprobes. In the meantime the Hirogen hunter escapes from Sickbay (deck 5) and soon faces off Seven and attacks 8472. When Seven beams them off the ship, Harry says that 'someone has tapped into transporter control on deck 5'.
Character mistake: B'Elanna replicates and assembles a television from 1956, including a remote control. Tom however says they didn't have remote controls in the 50's. The first wireless remote was invented in 1955 and the "Zenith Space Command" (a remote looking similar to the one in use) came out in 1956, which fits the timeline. Tom is vastly aware of 1950's technology (he seems to know when the television remote came out) and wouldn't make that mistake or B'Elanna would have corrected him given how much research she had to do to set the whole thing up.
Character mistake: Captain Janeway's main goal in this episode is to proceed to the holodeck and shut down the lightning shield protecting Chaotica's death ray so Lt. Paris can shoot it with his destructo beam. Throughout the scene where Lt. Paris further briefs Capt. Janeway on her way to the holodeck, the plot is such that her goal is to shut down the death ray, not the lightning shield.
Character mistake: As Voyager is bombarded by the cyclone, Paris reports that Voyager is at 20,000 meters and falling. 28 seconds later, he reports that Voyager is at 18,000 meters and states that, at this rate, they will crash in 10 minutes. However, if they drop 2000 meters in 28 seconds, they would actually crash in 252 seconds (just over 4 minutes). One might argue that perhaps it was calculated that their descent would slow down as they approached the surface, but later they fall from 2000 meters to 1000 meters in 14 seconds - the same rate.
Character mistake: When trying to trap the energy mass they created, B'Elanna announces, "Now it's trapped on three sides," to which Paris replies, "It's changing course. Heading for the open side." Since we function in 3-dimensional space, unless they are trying to trap it in a triangular-based pyramid, the number of "sides" to be concerned with is six.