Plot hole: Willow hacks into The Initiative's computer system and discovers air ducts and electrical conduits leading to Adam behind 314, but the room doesn't show up on the schematic. It seems unlikely that nobody who didn't know about Adam would have noticed this and investigated.
Plot hole: In 'Becoming, part 2' Spike decides he doesn't want to see the world end and makes an alliance with Buffy. Later on, however, he helps Angel get the information he need from Giles to end the world. If he's trying to save the world, why help Angel destroy it even faster? He didn't need to buy Giles any time - he had already convinced Angelus to keep Giles alive for the time being. If he had truly reached breaking point later on Spike could have called in Drusilla then, but as it is he helped Angelus way too early and nearly cost Buffy her life. [Some people disagree at great length, others don't - I haven't seen the episode myself, and the explanation of the mistake and the corrections just make my brain melt. File this under "make up your own mind and just let it go"].
Revealing mistake: When Buffy is fighting the vampires that are trying to resurrect The Master, at one point she is grabbed from behind by a male vampire and kicks a female vampire in a green top twice (right after Absalom shouts "The sacrifices! Stop them!"). If you watch closely, both of the kicks miss the vampire, revealing that the fight was choreographed. The punch that she delivers to the male vampire afterwards also looks fake.
Continuity mistake: After Doris from Social Services leaves Buffy's house and Spike leaves after her, he takes his lighter out of Buffy's pocket. But earlier in the episode when everyone was packing away all of Willow's magic stuff, Buffy finds Spike's lighter and puts it in one of the boxes.
Continuity mistake: When you first see Adam lying on the bed, before he wakes up you see his human side of his face on his right side. In the next scene the camera is over Maggie Walsh's shoulder and the human side of his face is now on the left side. Then just after he stabs Maggie it switches back again.
Visible crew/equipment: Near the end after freeing the prisoners, Buffy fends off Ken's henchmen on a raised platform. At some point during the fight she is grabbed by one of the henchmen and punched in the stomach by two others. She then breaks free and throws one of them off the raised platform - onto a large crash mat. Please note that this is a separate mistake to the one already submitted as it is talking about a different crash mat.