Revealing mistake: Early in the final fight, Vi makes a flying kick to knock an Ubervamp off the cliff. As she lands, she swings back and twists slightly, suspiciously like one would do if suspended on a wire. There is no way she could gain this motion by kicking straight off the vamp as shown.
Continuity mistake: Sunnydale apparently has docks ('Consequences' and 'Surprise') and is quite close to a beach ('Go Fish'), but when Sunnydale is destroyed in 'Chosen', we see an overhead shot of the crater and it appears to be in the middle of the desert, far from any coastline.
Continuity mistake: Towards the end of the episode when Buffy is running along the rooftops trying to catch up to the school bus, we see an overhead shot of one of the buildings collapsing just before she jumps off it. In this shot, Buffy is not holding the scythe that she used to kill Caleb. Once she jumps the rooftop however, she is holding it in her right hand.
Continuity mistake: When Buffy has told the Scoobies her plan about going down into the Hellmouth, Willow starts talking about how it's beyond anything she's ever done. As she says this she is looking in Buffy's direction, but as she continues and the camera cuts away, her reflection can be seen in a nearby mirror - she is suddenly looking straight ahead of her and then turns to look in Giles' direction. She is then suddenly looking back in Buffy's direction as the camera cuts back to her.
Continuity mistake: When Anya leads Andrew away to their station in the school, she goes from having her right arm over his shoulders to having her left arm over her shoulders between shots. She is off camera for a second, but not long enough to move to the opposite side of him and change hands.
Continuity mistake: When Buffy is giving her speech about being chosen, there is a shot of Andrew sitting on the couch next to Xander and Anya, but in several of the shots of the Potentials he can be seen standing on the other side of the room in front of the curtains. It's definitely him - his face isn't visible, but the clothes are the exact same.
Continuity mistake: In the very last episode, Chosen, in the shot right before Spike's amulet sends out the blast through the roof, you can see an ubervamp just about to pin him against the wall and in the next shot he is standing in the middle of the area backing up toward the stairs and then he sends out the blast. There would never be enough time between the shots for Spike to break free and then go and stand in the middle of the floor.