The difference one might see between Dostoevsky and Camus is that Dostoevsky ultimately concludes that we cannot live without faith, while Camus believes that we can. In Crime and Punishment, the protagonist, Raskolnikov, commits a murder in order to test the limits of his own freedom. He is subsequently wracked with guilt, ultimately confesses, and experiences a conversion in the epilogue. In The Brothers Karamazov, Ivan Karamazov's atheism ultimately leads him to madness, while his younger brother, Alyosha, who ardently wants to believe, emerges in better shape.
Raskolnikov, Ivan Karamazov, and Kirilov are different from most atheists in that they want to live consistently with their principles. It is not enough for them to assert that they are free and to continue to live as before. They must sort out exactly how a life without God would be different and try to live according to that rule. For Raskolnikov, this path leads to murder, for Ivan it leads to madness, and for Kirilov to suicide. Camus similarly wants his characters to fully live the philosophy he entertains. Meursault and Caligula, two of Camus's protagonists created around the same time that he wrote The Myth of Sisyphus, do not simply accept the absurdity of their lives on an intellectual level. Camus also uses them to see how a fully consistent absurd life might differ from the norm.
Much of this difference between Camus and Dostoevsky, however, might be explained by sixty or seventy years of history and a different cultural climate. They needn't be seen as contradictory. In Dostoevsky's Russia, life without God may have seemed impossible, whereas in Camus's France, life without God might even have seemed necessary. From Camus's discussion of The Possessed, he certainly seems to accept that suicide was the only alternative to faith in God at that time.
Camus includes this chapter because he wants to see whether a writer who accepts absurdist principles must necessarily remain true to those principles. In Dostoevsky, he seems to conclude that it is possible for a writer to recognize absurdity and not live according to that principle. In the first part of the Myth of Sisyphus, Camus shows that existentialist philosophers such as Jaspers, Kierkegaard, and Chestov recognized absurdist principles but then leapt into faith rather than choosing to accept those principles. In this chapter, he shows that what is true for those thinkers philosophically is also true for Dostoevsky as a writer. An absurd sensibility does not necessarily imply absurd fiction.