Usually, philosophy is advocacy. Sometimes it's disruption without a positive thesis in mind. More rarely, it's confession.
The aim of the confessional philosopher is not the same as that of someone who confesses to a spouse or priest, nor quite the same (though perhaps closer) as that of a confessional poet. It is rather this: to display oneself as a model of a certain sort of thinking, while not necessarily endorsing that style of thinking or the conclusions that flow from it. Confessional philosophy tends to center on skepticism and sin.
Consider, in Augustine's Confessions the famous discussion of stealing pears, wherein Augustine displays the sinful pattern of his youthful mind. Augustine's aim is not so much, it seems to me, to advocate a certain position (such as that sinful thoughts tend to take such-and-such a form) as to offer the episode for contemplation by others, with no pre-packaged conclusion, and perhaps also to induce humility in both the reader and himself. He offers an analysis of his motives -- that he was trying to simulate freedom by getting away with something forbidden (which would fit with his general theory of sin, that it involves trying to possess something that can only be given by god) -- but then he undercuts that analysis by noting that he would definitely not have stolen the pears alone. Was it then that he valued the camraderie of his sinful friends? He rejects that explanation also -- "that gang-mentality too was a nothing" -- and after waffling over various possibilities he concludes "It was a seduction of the mind hard to understand.... Who can unravel this most snarled, knotty tangle?" (4th c. CE/1997, p. 72-73)
Descartes's Meditations, especially the first two, are presented as confessional -- perhaps partly to display an actual pattern in his past thinking, but perhaps also partly as a pose. Here we see or seem to see the struggles and confusions of a man bent on finding a secure foundation for his thought. Hume's skeptical conclusion to Book One of his Treatise seems to me more genuinely confessional, when he asks how he can dare to "venture upon such bold enterprizes when beside those numberless infirmities peculiar to myself, I find so many which are common to human nature" (1739/1978, p. 265). "The intense view of these manifold contradictions and imperfections in human reason has so wrought upon me, and heated my brain, that I am ready to reject all belief and reasoning.... I dine, I play a game of back-gammon, I converse, and am merry with my friends; and when after three or four hours' amusement I wou'd return to these speculations, they appear so cold, and strain'd, and ridiculous, that I cannot find in my heart to enter them and farther (p. 268-269). We see how the skeptic writhes. Hume displays his pattern of skeptical thought, but offers no way out, nor chooses between embracing his skeptical arguments and rejecting them. Nonetheless, in books two and three he's back in the business of philosophical argumentation.
Generally, it's better to offer a tight, polished exposition or argument than to display one's thoughts, errors, and uncertainties. That partly explains the rarity of confessional philosophy. But sometimes, no model of error or uncertainty will serve better than oneself.
[for some discussion, see the comments section of the original post]