When Buffy sees the girl's body she is wearing heels, then, during the fight with the ubervampire, she wears sneakers, but when she kicks the pipes, she is wearing heels again.
When Andrew leads Buffy, Xander and Dawn to the Seal of Danzalthaar in the school basement, at 07:41 as Dawn approaches the large wheel, her hair goes from curled to straight then back again.
Spike is shown being tortured by drowning - that is, his head is held down in water and his breath bubbles up to the top - but according to Joss Whedon's canon, vampires do not breathe, or at least, they do so only for effect. This "vampire fact" was shown in Season One's Angel (1997), when Angel cannot perform CPR on Buffy, but Xander can. Meanwhile, in Season Four, Spike fakes heavy breathing as though he has been running, but it's just a trick so that he can get into Giles's house. (That said, it is possible that there are some instances where vampires require breath - specifically, when they are in need of healing. This occurs sporadically throughout the series - when Spike is being physically tortured, he seems to need air to heal himself.)
Most of the time, when Xander is sweeping at the beginning, the broom isn't touching the ground.
When Buffy and Giles are investigating the tree farm, there is a random man in what is supposed to be an empty lot.
Buffy states that if the First does kill off the potential Slayers and their Watchers, they'll then come for Faith, then her. This is incorrect. Faith, who was called when Kendra died, is the current "active" Slayer, meaning they would go for Buffy before Faith.