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During every Shemoneh Esrei of mussaf, we pray that the Beis Hamikdash be rebuilt. However, only on Yom Tov do we mention that it was destroyed due to our sins - ומפני חטאינו. On Shabbos, and on Rosh Chodesh, we don't mention this.
However, when Shabbos and Rosh Chodesh coincide, we do mention our sins - ולפי שחטאנו.
If neither Shabbos nor Rosh Chodesh demands a mention of our sins, why does the combination require it?
In the Kuntras Yekar Tifferes (http://beta.hebrewbooks.org/pdfpager.aspx?req=36072&st=&pgnum=204) the author brings this question in the name of HaRav Zvi Perzovicz shlitoh who provides an interesting answer. The Tur in the beginning of Hilchos Rosh Chodesh says that really the 12 Roshei Chadashim should have been Yomim Toivim, however, because of the Cheit Ha’eigel, Rosh Chodesh was turned from a Yom Tov to an almost ordinary day of the week. The Shulchan Aruch says that you should be Marbe K’tzas Simcha on Rosh Chodesh. However, it does not have a Din of Yom Tov. The women who did not sin by the Cheit Ha’eigel have a very special obligation and Mitzvah to treat Rosh Chodesh as a bit of a feeling as a Yom Tov. He suggests that on Shabbos Rosh Chodesh we retain the original aspect of Yom Tov. Since Shabbos is Mai’ain Oilam Haba, and is L’maila Min Hacheit, when Rosh Chodesh falls out on a Shabbos it has a higher level of Kedusha like a Yom Tov. Hence the change of nusach. See there for several proofs to this idea
