Audio problem: The drummer in Oz's band's movements don't match the song that is heard when we first see them performing at Buffy's house.
Continuity mistake: When Buffy decides to leave again and goes to her room to pack, her door is three quarters shut when she goes in, half open when Willow comes in and almost fully open when Joyce comes in. At no point does anybody touch this door.
Character mistake: When the zombies break into Buffy's house, she stakes one to check if it's a vampire. But she should know full well that it isn't a vampire as they all broke into her house without being invited.
Continuity mistake: When Pat puts on the demon mask and punches Buffy into the bedroom door, she hits the door with her back and then falls to the floor on her face. In the immediate shot she hits the floor again, but this time on her back.
Continuity mistake: When Joyce throws the bag with the stray cat into it into the hole, the bag lands with the knot at the top right, and shows no signs of moving. They chat for a few seconds about something to say, then when Buffy starts filling in the hole (with no sound of a moving bag), the bag's now lying with the knot on the left.
Continuity mistake: After Buffy is supposed to meet Willow and she is walking home you can see she is wearing light blue wedge heels but when she is down in the basement she is wearing completely different purple heels.
Answer: She has superpowers, ergo fast movement. The reason she outruns him so easily in "Touched" is that she stops trying to actually fight him and just dodges instead.
Shay