Revealing mistake: When Gwendolyn Post fights Angel in the mansion, she breaks a long-handled shovel over her knee to create a stake. If you watch in slow motion, the handle of the shovel is pre-broken (she simply holds the two pieces together tightly), and the pieces separate just before they actually make contact with her knee.
Other mistake: When Buffy and Willow find the picture of the noblewoman in the Watcher's Diaries, Buffy asks "Who is she?" and Willow replies "It doesn't say", but if you watch closely you can see that during the brief shot of the book her name is visible next to her picture.
Continuity mistake: In the scene when Spike was in Buffy's kitchen with Joyce, Buffy keeps him down on the table and grips Spike by the throat. She uses her left hand (in the close shot it is also the left hand) but in the next scene when Angel is also in the kitchen and Spike hits Buffy's hand, she uses her right hand (Spike hits her right hand).
Continuity mistake: In Innocence, when the Scoobies are in the library, Oz (I think) says for the crew to meet outside Xander's house a half hour later. Then we cut to Buffy strolling through Sunnydale High in broad daylight, then back to Oz's van outside the military base and it's last night again.
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: In 'Tabula Rasa' when the loan shark's henchmen throw a stake at Spike at the start it flies over his head and hits a tree. But, if you compare the angle they approach from with the relative positions of Spike and the tree you can see it would have had to do a 90 degree turn in mid air to fly over his head before hitting the tree.