Deliberate mistake: Every time the commandos are seen in the early episodes of this season (The Freshman and Fear, Itself) they wear ski masks to protect their identities. Starting in The Initiative, they stop doing this. The reason is that the audience needs to be able to identify the different commando characters, but it's still a discontinuity.
Deliberate mistake: After Jenny pushes the janitor's cart into Angelus and runs up the stairs, there's no physical way he could be waiting for her at the top of the stairs (even with vampiric speed). Joss Whedon has admitted in interviews that he put in this blatant continuity error as the situation is a staple of the horror genre (girl runs from villain, only to find him standing right in front of her).
Deliberate mistake: Though vampires are said to possess super-human strength, and are in fact demonstrated to have it, they are often kept confined by the flimsiest of restraints. One distinct example is in season 2 episode 9 when Kendra locks up Angel he is unable to break out despite being held in by nothing but a flimsy padlock. This mistake is pointed out in the DVD commentary as an unavoidable 'mistake'.
Deliberate mistake: In the fighting scene, the glove of Myhnegon is used by the new watcher who directs it toward the ceiling of Angel's living quarters where there is a huge skylight. The skylight is not seen in any other episode and its presence would have burned him on other occasions. (00:38:10)