When Angel, Willow and Xander are talking to Buffy and Owen, Buffy's height constantly changes between shots. Sometimes she's standing on a platform (so she can fit in the same shot as the super-tall Owen), sometimes she's not.
When the vampire breaks the glass with his head, the remaining glass disappears.
At the end, Owen is trying to set up another date with Buffy. In the background, a woman in a red and black jacket carrying a red bag walks up the stairs. Seconds later, in the next shot of Owen, the same woman is walking up the stairs without the bag. Seconds after that, in the next shot of Buffy, the same woman with the same red bag on her shoulder is walking by at ground level.
Two different actors were used for playing the part of the driver of the bus that crashes. The one that speaks to the passengers is bald, but the one seen at the wheel when the vampire stands in the road causing the crash, has long hair.
In the famous scene where Buffy tells Giles, "If the Apocalypse comes, beep me," she grabs her pager from out of shot. Trouble is, she's standing in the middle of the hall and there's nowhere from where she could have picked up the gadget.
At the funeral home, when Buffy and Giles open up the different drawers to look for the Brethren, they open the same drawer every time. The crew made a few changes in the background and Buffy is probably on a platform the second time she looks, but it's the same. You can tell by the smudges on the inside of the door.
The driver of the shuttle bus is bald but during the exterior shots of the bus before it hits a vampire, the stunt driver is seen to have long hair.
At the funeral home, when the new vampire awakens, his reflection can be seen in the glass cabinet doors behind him. Part of vampire lore is that they aren't supposed to reflect in mirrors, or any glass surface.