Continuity mistake: A cut scene shows the sun had just gone down right before Buffy heads to the Bronze and Joyce tells Buffy she will make them dinner if she chooses to come downstairs. This would assume it is evening. When Buffy kills Luke she says sunrise is in 9 hours. Considering this episode takes place in March, that would mean the sun would have gone down around 7pm. But 9 hours from sunrise would be around 10pm.
Continuity mistake: When Buffy is shown pulling the secret chest out of her closet with weapons against the vampires, the window behind her shows daylight, but when she steps out of her window it is fully dark. There is also a cut scene right before that shows it was twilight, but there was late afternoon sunlight shining through the windows of her room. It's not possible for the sun to go down that fast.
Visible crew/equipment: When Buffy is going through the murdered Professor's office, Angel seems to trip through the doorway. To the top right above Angel's head a boom mike is visible twice.
Visible crew/equipment: When Buffy and Riley are talking at the crime scene, where he tells her about Maggie Walsh's death, one of the camera shots has a very visible black border around the entire edge of the screen as if the lens shade was visible in front of the lens and could be seen in the shot. (00:18:20 - 00:19:40)
Visible crew/equipment: After Buffy walks outside to hear the sirens, in the close up of Buffy's face on the upper right corner of the screen you can see a boom mike. (00:06:30)
Plot hole: Angel says he cannot give Buffy mouth-to-mouth because vampires have "no breath." However, throughout the entire series, vampires are shown to breathe, and are able to do things like smoke, gasp, get choked out, etc. (Granted, the franchise also confirms a vampire cannot die from asphyxiation, but still they're shown to breathe regularly).
Other mistake: In the very first shot (a long tracking shot backstage), watch the girl practising with her tuba. At one point she pulls her mouth away and then leans back in and starts playing some notes. But you can tell that she isn't blowing into it, and in fact, her mouth isn't even quite touching it. So it shouldn't be producing any notes. Then, almost immediately after, you can also see the actress purposely moving the tuba (sort-of rotating it to her right) to get out of the cameraman's way.
Factual error: When Willow is chatting with "Malcolm," she says she has to sign off the chat program after she gets nervous. However, all she does to "sign off" of the chat program is turn off the computer monitor. This would do literally nothing. The program would still be running if you turned the monitor back on. Yet the episode treats it like she's just turned the entire computer off by turning off the monitor.
Continuity mistake: Anya's hair changes from straight in the previous episode to curled in this one, even though it is set directly after.
Nightmares - S1-E10
Continuity mistake: Buffy's history teacher is not the same person from episode 1 or 9, originally a woman, the teacher is now a man. Cordelia also notes Buffy doesn't know where the class is due to ditching, but we saw Buffy in her history class with Cordelia in The Puppet Show. How could Buffy not know where her class is only 1-2 weeks later?
Other mistake: Buffy freaks out when she's first taken to the hospital. Joyce tells the gang it was because her cousin died when they were young, but in season one episode "Angel" Buffy has no problem being in the hospital, same with later seasons.
Welcome To The Hellmouth (1) - S1-E1
Plot hole: There was no need for the master wait 60 years for the night of the Harvest, when all he had to do was have his minions kidnap and gather a bunch of humans and bring them down to where he is so he could feed on them himself and escape 60 years earlier.
Chosen answer: Angel is saying that he is still jealous and Buffy recognizes that this is his way of saying he wishes that they could be together. That is why they both smile.