Factual error: The story takes place in 1957 (Sputnik is already in orbit). At the end, T'Mir sells her invention, which ends up being Velcro (technically hook and loop fastener). However, George de Mestral obtained the patent for Velcro in 1955, and by 1957 he had already expanded into the United States.
Continuity mistake: When Reed instructs Archer how to disarm the second arming mechanism of the mine, he tells him to turn the first component clockwise, but then Archer is shown turning it counter-clockwise. Then Reed tells him to turn the rest of the components counter-clockwise, but afterwards he must reverse the last component to keep the mine from rearming, which means turning it clockwise. But when Archer reverses the last component, he turns it counter-clockwise.
Continuity mistake: When Captain Archer is talking to Hoshi in her quarters, at one point he leans against the door-frame. In the next shot of him from that angle, he's no longer leaning, even though we saw him from behind and he didn't move. In the next shot, he's back to leaning against the door-frame.
Other mistake: Just as Malcolm finishes opening the hatch on the alien pod, a graphical mistake appears towards the left side of T'pol's arm. It appears as if a section of the craft had been added with an overlay, which abruptly disappears as the cutting tool stops firing. (00:01:30)
Factual error: When Capt. Archer joins Travis in a weightless environment he refers to in an earlier episode as the ship's "sweet spot", Travis discusses the bad news he received from The Horizon, his parents' cargo ship, with Capt. Archer, and you see tears rolling down (and off) Travis's face. In a weightless environment the tears would not react in this way. (00:06:20 - 00:07:40)
Other mistake: Two Borg are discovered North of the Arctic Circle. One is covered in ice, the other encased in ice. Carbon dating shows the wreckage to be over 100 years old (to tie it in with Star Trek:First Contact). The Borg are inactive and unable to repair themselves until they are "defrosted". The temperature above the Arctic Circle doesn't usually drop below -100°F. The Borg have been shown to work very well in the vacuum of space which is 3-4K (roughly -454°F). If they can survive that, even being frozen in a solid block of ice would not slow them down. The nanites have to be in their "blood" and operate at the cellular level. That cannot be damaged by re-entry and the crash. [This is one of those entries that keeps being corrected and recorrected - I'll leave it up to people to make up their own minds.]