The Shatarudra Samhita gives two reasons in one breath. Brahma has told the lie about the pillar of light, and he has begun to think of himself as Shiva's equal because he too has five heads. Shiva's answer is Bhairava, a form drawn out of himself, who removes the offending head. The skull will not leave Bhairava's hand and the sin of brahmin-killing walks behind him. He roams the three worlds until, at Kashi, the sin falls away and the skull drops. The ground where it lands becomes a place of release.
Kashi keeps the ending. At the Kapalamochana tirtha near the Kaal Bhairav temple the skull of Brahma is said to have fallen from Bhairava's hand, and the Kashi Khanda of the Skanda Purana promises that merely remembering the spot removes sins. Bhairava stays on as the city's magistrate.
What is different here:
Localises the release at a named tirtha and adds the merit of remembering it.
The Matsya Purana makes Shiva himself, not an emanation, the one who incurs the sin. The skull clings to his palm and he must atone as Bhikshatana, the naked wanderer begging with it as his bowl, until the journey ends at Varanasi and the skull falls.
What is different here:
Shiva rather than Bhairava does the deed; the atonement is the Bhikshatana wandering.
In the Kurma Purana the offence is spoken pride: before an assembly of sages Brahma declares himself the supreme creator. Shiva takes the fifth head, then begs with the skull until Varanasi, where it drops at the spot named Kapalamochana. There is no ketaki flower in this account.
What is different here:
Trigger is a boast at a council; no pillar of fire and no ketaki.
The Skanda Purana ties the beheading to the daughter-desire story: Brahma's fifth head is cut because he committed incest with his own daughter, so the punishment that the Vedic gods once gave Prajapati through Rudra is given to Brahma through Shiva.
What is different here:
Trigger is the incest motif; links this story to brahma-and-saraswati.
In the Varaha Purana Brahma loses the head for calling his own son a Kapali, a skull-bearer, a taunt aimed at Shiva who is born as Brahma's son in that text.
The Tamil poet-saints of the seventh and eighth centuries, Campantar, Appar and Cuntarar, sing Bhikshatana in a gentler key: the wandering beggar whose beauty makes the sages' wives forget themselves and press alms on him. The skull is still there, but the mood is love rather than penance, and Chidambaram carries the image in festival procession.
What is different here:
Emphasis moves from the sin to the beggar's charm and the women who give him alms.