Deliberate mistake: After Jenny pushes the janitor's cart into Angelus and runs up the stairs, there's no physical way he could be waiting for her at the top of the stairs (even with vampiric speed). Joss Whedon has admitted in interviews that he put in this blatant continuity error as the situation is a staple of the horror genre (girl runs from villain, only to find him standing right in front of her).
Continuity mistake: Huge continuity gaffe. When Anya and Xander see Spike and the robot in the graveyard it is nightime. Xander goes back to speak to Spike almost immediately and sees Glory's followers kidnapping him. He wakes goes back to Buffy's house and from there they, again almost immediately, go looking for Glory's house - and it is the broadest of broad daylight. You could see it was still night before they left through the windows of the house. It also begs the question of how they got Spike from Glory's house back to his crypt in the sunlight since they had no blanket.
Visible crew/equipment: When Buffy is searching through the backstage area (before the chandelier falls on her), there is a mirror in the background. At one point in the mirror, you can see the reflection of a crew member. Pause for a better look but it can be seen at normal speed.
Continuity mistake: Spike tells Buffy about drugs the Initiative would give him when his chip started firing after he was first captured. This is wrong in two ways: firstly, we saw most of Spike's time in captivity, and he escaped before they could drug him, and secondly, he didn't even know he had a chip until he escaped and tried to kill Willow.
Factual error: There is a short picture of a 17th century German newspaper. The actual text however not only is written in modern High German as opposed to Mid-High German, but also is full of errors (mixing female articles with male nouns is only one example). Funnily enough, Giles' incantation at the end of the episode is in correct German.