Deliberate mistake: Every time the commandos are seen in the early episodes of this season (The Freshman and Fear, Itself) they wear ski masks to protect their identities. Starting in The Initiative, they stop doing this. The reason is that the audience needs to be able to identify the different commando characters, but it's still a discontinuity.
Continuity mistake: When Buffy is fighting the vampires at the end of the episode, she grabs a tennis racket and smashes the brunette vampire over the head. She falls over the back of the couch and we hear the thud of her landing, however in the wide shot of the couch the vampire has just started to go over the back.
Continuity mistake: In the scene where Buffy and Kathy are deliberately infuriating each other (tapping pencils and cracking eggs) there are several lights on throughout the room (on Buffy's desk, above Kathy's computer, in the kitchenette) yet when Buffy turns out her bedside lamp, the entire area is thrown into darkness.
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.
Continuity mistake: When Buffy and Spike are fighting on campus during the day, Xander tries to help Buffy but Spike throws him against a lamppost. When he lands in the ground he falls completely on the grass on his side but if you look behind Spike in the next shot he lies half on and half on the grass on his front.
Continuity mistake: When Xander is showing Willow and Buffy his fake ID and the view of the shot is from behind Xander, he is holding the card with his thumb and middle finger. Switch to the view from behind Willow and it's being held with his thumb and index finger. Then it switches to behind Xander again and it's back to thumb and middle finger.
Continuity mistake: When Veruca comes to Oz's cage to spend the night there, she tosses her jacket onto the round stone thingie (knob?) at the end of the stairs, and it covers the whole thing. However, when Willow discovers them in the morning and Veruca goes to pick up her jacket, you can see that it has moved. Willow didn't touch it when she entered, and both Oz and Veruca were locked up so they couldn't get anywhere near it. (00:26:35 - 00:29:20)