Deliberate mistake: When Xander ducks under the caution tape to catch up to the group of students at the beginning, as the shot is about to cut, judging by his casual walking pace, he should maybe be 10 or so feet ahead of Buffy and Willow. When the camera cuts, he's suddenly at least 30 feet away. It's probably edited that way for time reasons (to quicken the pace of the scene rather than show him walking an extra 10 seconds), but still a minor mistake.
Deliberate mistake: When Willow goes into her dark bedroom, she turns on her lamp. Somehow, her tiny lamp illuminates the entire room and also casts a shadow behind her, even though it's behind her. Clearly, when she turns on the lamp, they also turned on some set-lights in front of her to make the room bright enough for filming. There's also something else that gives it away-when she turns on the lamp, the studio lights come on a split-second later - just enough that if you look closely, you'll notice the delay.
Deliberate mistake: When Buffy, Xander and Giles try to "delete" Moloch from the computer, watch the computer screen as she sits down. There's already a folder labeled "Willow" on the desktop. Buffy then types in the word "Willow" to search for the document (without clicking on any sort of search bar, so typing it would do nothing), then says something along the lines of "That's probably it" and proceeds to click on the "Willow" folder... that was already on the desktop. The entire scene makes no sense.
Continuity mistake: When Buffy and Willow arrive at school at the start of the episode, Willow is wearing very dark, completely opaque tights that look to be black or at best very dark navy blue. When Xander finds her in the next scene, which takes shortly after on the same day, she's suddenly wearing much brighter purple tights.
Continuity mistake: In the beginning of the episode, when Willow gets the email from "Malcolm" and says "He's so sweet! What should I write back?", she instantly goes from facing forward at the computer to facing sideways at Buffy between cuts.
Welcome To The Hellmouth (1) - S1-E1
Continuity mistake: When the Master first starts to appear, rising out of the pool, the top of his head is wet. But in the next shot (a wideshot) where you see him continue to rise, he's bone dry. This is because in the first shot, the actor is really coming out of the liquid, whereas in the next shot, he's added through visual effects. He obviously was supposed to look "dry" the entire time, but because the first shot was done with an actor rising out of the pool on-set, there was no way for him to emerge dry.
Welcome To The Hellmouth (1) - S1-E1
Continuity mistake: After Cordelia turns down Jesse at the Bronze, he says "Fine plenty other fish in the sea." In the first shot, his arm is raised and he has his hand against a pillar. In the second shot, his arm is suddenly down by his side.
Welcome To The Hellmouth (1) - S1-E1
Continuity mistake: When Jesse asks Cordelia if she wants to dance at the Bronze, and she replies "With you?", she goes from facing downwards to facing sideways towards Jesse between cuts. They try to hide the continuity mistake with editing (cutting the shot as an extra is partially obscuring the frame as he walks by), but you can see clear as day that she goes from looking down to looking sideways instantly.
Welcome To The Hellmouth (1) - S1-E1
Continuity mistake: When Buffy meets Willow at the Bronze, a drink basically magically materializes in Buffy's hands at one point. She points to the bartender like she wants a drink, but a few shots later (when she says "How come?") she suddenly has a drink in her hand without having picked it up in any previous shot. She also goes from not sipping her drink to sipping it between cuts just a few shots later.
Welcome To The Hellmouth (1) - S1-E1
Continuity mistake: When Buffy meets Willow at the Bronze, when Willow says she thought Xander was going to be there, Buffy is standing about a foot away from the bar and her arms are by her side. But suddenly in the next shot, she's right up against the bar and leaning on it with her elbow. Then in the next shot, she's about a foot away again.
Welcome To The Hellmouth (1) - S1-E1
Audio problem: When Buffy meets Angel, when she says the line "What I want is to be left alone!" and Angel replies "You really think that's an option anymore?" pay attention. Buffy's last few words don't quite match her lip movements, and Angel's lips don't seem to be moving at all for his line. The audio looping is a little sloppy. This is better seen in the brighter HD remaster, but is also noticeable in the original, darker standard definition master.
Welcome To The Hellmouth (1) - S1-E1
Continuity mistake: This is hard to describe. When Buffy and Joyce are talking in her room, Joyce keeps pulling pillows off Buffy's bed and re-arranging them. However, at one point, when the shot cuts back to her, she's holding a different pillow than in her last shot. This technically wouldn't be a mistake since she's off camera except in the shots facing Buffy, there's a mirror behind her with her bed in view, so we should have Joyce put down the first pillow and pick up the second one... and we don't.
Welcome To The Hellmouth (1) - S1-E1
Revealing mistake: When Buffy sneaks into the locker room and checks out the dead body, you can see part of the outline around the prosthetic vampire bite on his neck (it isn't quite blended in properly), revealing it's an appliance.
Welcome To The Hellmouth (1) - S1-E1
Visible crew/equipment: When Buffy's mom drops her off at school at the beginning, watch closely when Buffy gets out of the car. In the passenger door's quarter glass window, you can see a reflection of a set-light (probably a fill light to eliminate shadows) for a few seconds when she opens the door.
Welcome To The Hellmouth (1) - S1-E1
Visible crew/equipment: In the beginning, when Buffy is having the prophetic dream, there's a shot right as it's about to end where the camera is above her and moving down closer to her. If you watch, you can blatantly see a shadow (more than likely the camera rig) grow over her as it gets closer and closer to her.
Out Of Mind, Out Of Sight - S1-E11
Visible crew/equipment: When Buffy and Cordelia enter the mop closet you can see the shadow of the boom microphone on the wall near Buffy's head (left side of screen). (00:31:44)
Out Of Mind, Out Of Sight - S1-E11
Continuity mistake: When Cordelia is giving her May Queen speech, Willow notices 2 body guards standing around. In both shots you can see the same 2 girls, wearing light blue and grey, walk behind The Body guards. The same girl in light blue can also be seen in several different shots, in the crowd listening to Cordelia and walking around as Buffy talks to Willow and Xander.
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.
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).
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.