The Yerushalmi (Shavuos 6b in Vilna ed) writes that when Moshe heard that the viduy of the sa'ir l'chatas is mechapeir he recited the chapter of tehillim "mizmor l'todah". It seems that the Yerushalmi understands the term "todah" in "mizmor l'todah" as an expression of viduy, contrary to the views of poskim (see Tzion Yerushalayim on the spot) who read it as an an expression of thanks, hoda'ah.
Perhaps there is a way to reconcile the two views. R' Yosef Engel in his derasha for shabbos shuva cites a Tosefta which tells of a man who instructed his son to make an elaborate seudas hoda'ah because he fulfilled the mitzvah of shikcha. The father explained that being able to fulfill shikcha gave him more joy than any other mitzvah because shikcha occurs only where a person forgets crops in the field unintentionally. Hashem brings about the mitzvah without the doer intending it and gives him the credit! Therefore, a special hoda'ah is required.
The same holds true of tshuvah. Tshuvah m'ahavah transforms all sins into merits, yet it is obviously prohibited to sin in with the intent to then do tshuvah to accrue more merits that way. Like shikcha, these zechuyos come without the person intending to do good (in fact, the person had just the opposite intention), yet Hashem gives credit anyway. Therefore, like shikcha, these zechuyos which come through our viduy and tshuvah warrant a special hoda'ah to Hashem.
Interestingly, the MG"A (O.C. 51:10) quotes a minhag (which we do not have) to recite mizmor l'todah on R"H and Y"K. He attributes the minhag to the fact that the mizmor contains the words "hari'u l'Hashem", hinting to tekiyas shofar. Based on the Yerushalmi perhaps the minhag is rooted in the relationship between the today and viduy.
(Also see Sefas Emes, Chanukah leil 3, 5631, who explains hoda'ah in a way that relates to viduy.)