Audio problem: When Xander, Willow and Ampata meet up at The Bronze, the song the band is playing is at the point of a guitar solo. If you watch Oz (who has already been established as 'lead guitarist') on stage in the background; he's blatantly NOT playing that solo. The hand movements don't match up.
Continuity mistake: When the female watcher is being destroyed after Buffy cuts off her arm. Her sleeve is empty, which is not possible, as Buffy could not have removed the arm but not the sleeve. Also the arm on the floor is removed at the elbow, but the watcher has lost the entire arm in a previous shot.
Other mistake: Just after Giles becomes completely blind, Spike gets up to get a book off of the mantle. When he does this you can, very briefly, see James Marsters reflected in the glass doors of the bookcase that was behind him, despite Spike being a vampire.
Continuity mistake: When Buffy is walking forwards through Angel's dust in 'The Wish' there's a vampire running towards her, only a few steps behind. Then there's a shot of her walking forwards taken from another angle, then when we can see behind her again the same vampire hasn't moved forwards at all, he's still running from 3 steps behind her.
Continuity mistake: When Spike finally gets into the area where the gem is and Harmony follows him up there, he puts on that necklace thinking it's the gem. When he realizes Harmony has the gem it shows him taking off the necklace. It cuts to another shot and then shows him again with the necklace, but minus the charm.