The Temple of Set exists because of a disagreement about whether Satan is a metaphor.
Michael Aquino, an officer in the United States Army and a priest of the Church of Satan, left it in 1975 with a group of others. His objections were partly organisational (LaVey had abolished the local groups and was, in Aquino's account, selling priesthoods), and partly theological. Aquino held that the adversary was a real being, that the materialism of the Church of Satan led nowhere, and that the entity in question was better identified with Set, the Egyptian god of the desert, of disorder and of separateness, than with a figure out of Christian demonology.
That makes the Temple of Set theistic, which distinguishes it from both the Church of Satan and The Satanic Temple, and it is why it belongs in this section rather than outside it.
Its doctrine centres on Xeper, an Egyptian word it renders as I have come into being: the deliberate development of the individual consciousness into something separate from and independent of the natural order. Set is understood as the principle of that isolate intelligence, the one thing in the universe that is not simply a part of the universe, and the practitioner's aim is to become likewise. The Temple calls this the Left-Hand Path and defines it precisely as the pursuit of separation rather than of union, which is the reverse of what most religious systems ask for.
The break was amicable in neither direction, and the Church of Satan has treated every subsequent organisation calling itself Satanist as illegitimate ever since.