Vampires can't enter a house unless invited. Throughout Buffy, this appears to be some kind of force field, as if the vampire has run into a brick wall. But the vampire chasing Buffy and Angel gets its arm through Buffy's front door.
While visiting Joyce at the hospital, Buffy tells Xander was stupid because she let Angel into her house even after she knew who he was. Angel was actually invited in before Buffy discovered he was a vampire.
During the climax at the Bronze, Buffy puts down her crossbow and walks towards Angel, about five steps. When Darla bursts in a few seconds later, the crossbow is back at her feet.
When Buffy is walking away from Willow and Xander to leave the Bronze, you see her holding a dark jacket right after she feels Angel's presence behind her. As she is walking down the street, you can see she is wearing a dark jacket, either dark brown or black. The following scene, she is still walking but wearing a light baby blue colored jacket.
When Buffy leaves the Bronze (right before the Three attack her) to go home, she passes a strange-looking green-lit window. After the attack, she and Angel run past this window again, even though they're heading in the same direction Buffy was going anyway.
Towards the end, Darla fires over 40 rounds from two semi-auto pistols without reloading. The pistols used were single-stacked, using standard magazines, and couldn't possibly hold 20+ rounds each.
When the Three first attack Buffy, she and Angel rush back to her house and one of the Three manages to put his hand through her door, crossing the threshold. As they have not been invited, this could not have happened.
Buffy's scream and Joyce's scream are completely identical. The same scream was obviously used for both of them.
Near the end, when Buffy enters The Bronze, it is closed. As she gets to the bottom of the steps, the shot changes and as it changes, a person is visible hiding behind the pool table. The person's foot pops out on the right.
Angel says that he hasn't fed on a living human since the day he was cursed with a soul. But, we find out later, in Angel (1999), that he fed off of criminals for a while to try to pretend like he was still evil. He even turned a man in a flashback to WWII. However, it is possible that Angel just didn't want to get into an explanation about this.
Darla opens Angel's fridge to reveal it is full of donor blood packs. Later on in the series however, it is established that Angel does not consume any human blood whatsoever to avoid the temptation, living on the blood of butchered animals instead.