At the Career Fair, when the "Police Woman" calls Buffy's name and Buffy holds up her right hand to acknowledge, she suddenly has a small white bandage covering the tip of her middle finger (it was not there a few seconds earlier when she was talking with Willow and writing her name on the Career Fair sign-up sheet). It remains through the subsequent scenes in the library and Willy's Place, then vanishes for the fight in the church.
In the library Xander closes a book and squashes a bug in between its pages. In close up the book is gray in color with the title biology. Then in long shot it has a plain brown cover.
The length of Kendra's pendant chain changes. When she leaves, her chain is much shorter than it had been.
When Angel is rescued from Drusilla's torture, Angel is seen to be wearing Buffy's cross necklace (presumably a trade for his leather jacket, which she has worn several times already by now), yet as a vampire he shouldn't be able to tolerate a cross touching his skin.
Buffy is seen using a "steel" to sharpen her knife, while talking to Kendra. Buffy puts the knife down and uses the steel to whittle her stake but after a quick cutaway, the knife is back in her hand in the next shot.
Buffy, using a sharpening steel to sharpen her knife, is using it incorrectly. The manner in which she's using it would dull the blade in just a few strokes.
When Buffy, Giles, and company are contemplating what Spike has in store for Angel, Giles states Angel must killed under the light of the "New Moon". A "New Moon" is when the moon is not visible and at the beginning of it's waxing phase.
Immediately after Willy the Snitch turns Angel over to Spike, the view cuts to an exterior "stock" shot of Buffy's house. The house's real-life "1313" address, located on the porch support post to the left of the front steps, is covered up, but the hanging plant that is usually dressed in to obscure it is missing. Also, Buffy's "1630" address, normally dressed onto the porch support post to the right of the front steps, is missing in this shot, and a "1313" placard, usually removed or obscured, has been left hanging in plain view above the front steps. (Later, when Xander and Cordelia come running out, the "1630" is in its usual spot on the right-hand post and the hanging plant is in its usual spot obscuring the left-hand post; the placard is never in the shot.)
At 37:12 when Willow stakes the vampire Giles is holding, the vampire actor can briefly be seen crawling away out of camera view.
Act II opens with Giles, Buffy, Kendra and Willow doing a walk-and-talk in the courtyard of Torrance (Sunnydale) High. As they pass through a set of propped-open doors into a school building, a grip's spring clamp is visible at the upper left corner of the door jamb, holding a power cord in place. The cord loops up out of frame, likely to a "top light" rigged about three steps into the corridor, judging from the bright highlights that appear in everyone's hair as they pass underneath it.
Immediately after Willy drops him in the sewer, Angel - who, being a vampire, need not breathe - is seen panting.
At 25:10, Buffy puts down the knife she was sharpening, picks up a stake, and starts trying to whittle it with the knife sharpener.
Angel becomes terribly weak after Kendra locks him in the cage, but there is no reason why. The sun has not risen enough to harm him in any way, and his wound from the Taraka assassin has already healed.