This is a very interesting question and I will attempt to answer it to the best of my ability.

(1) (a) I saw in the Sidur Otzar ha'Tefilos - commentary "Iyun Tefilah" - (in the section on Musaf Rosh Chodesh) an explanation according to which it may be possible to answer your qusetion. He points out that the next few words of the Tefilas Musaf of Rosh Chodesh are "Biheyosam Makrivim Lefanecha Zivchei Ratzon". This teaches us that Rosh Chodesh only gives us a Kaparah when we are able to bring the Karbonos. This is the difference between Yom Kippur and between Rosh Chodesh. See Tosfos below 13b (at top of page) that Yom Kippur atones entirely even without the goat offering. This special quality of Yom Kippur is something which no other day in the year possesses and therefore even nowadays, when we do not possess the Beis Ha'mikdash, Yom Kippur can still provide us with a complete atonement (See Bi'ur ha'Gra Orach Chaim end #607).

[ It may be pointed out that the opinion of Rebbe (cited in Gemara Yoma 85b) is that Yom Kippur provides Kaparah even without Teshuvah. Even though the Halachah does not follow Rebbe, one can still say that for someone who did Teshuvah, the special nature of Yom Kippur gives Kaparah even without Karbonos.]

(b) The difference between Rosh Chodesh and Regalim can be seen in what we say in the respective Tefilos. On Yom Tov we say that they were given to us "le'Simchah u'LeSason" and we do not refer to Kaparah in the way that we do on Rosh Chodesh. So even though the Mishnah mentions Rosh Chodesh and Regalim together with regard to the fact that they both possess sacrifices which give Kaparah and this teaches us that there is no Kaparah without the Chatas, nevertheless there is a difference between Rosh Chodesh and between Regalim. The principal purpose of Rosh Chodesh is atonement, whereas the principal purpose of Regalim is rejoicing and the Kaparah provided by it's Karbonos are merely a secondary consideration.

(2) (a) According to the Otzar HaTefilos it follows that nowadays that we do not have the Beis Ha'mikdash, Rosh Chodesh does not give us Kaparah. However the Mishnah Berurah does not Pasken this way, because in the Halachos of Rosh Chodesh Orach Chayim 417:4 he cites the Pri Chadash in the name of Rabbi Moshe Cordovero (the author of Tomer Devorah) that the day before Rosh Chodesh is called Yom Kippur Katan because the sins of the entire month are forgiven then. One sees from this that nowadays also, Rosh Chodesh gives us a Kaparah.

(b) I would like to suggest sources that show how the main purpose of Rosh Chodesh is for Kaparah. See the Gemara below 9a which states that the reason that the Torah (Bamidbar 28:15) says that the Chatas of Rosh Chodesh is for Hash-m is because it is a special Kaparah since Hash-m made the moon smaller than the sun. The Rif (1b in Rif pages) writes that Hash-m repayed the moon for the fact that He had made him smaller than the sun, by giving the moon the honor that Yisrael are able to offer a Korban on Rosh Chodesh.

(c) See also Tosefta Shevuos 1:2 which cites Rabbi Shimon that the sin of making the Beis Ha'mikdash Tamai is more severe than all transgressions in the Torah because it requires 32 goat-offerings to receive a Kaparah. The time specified by the Torah for this Kaparah is Rosh Chodesh. We learn from this that Rosh Chodesh possesses a special quality of Kaparah to atone for the heaviest sin.

(d) My suggestion is that even though nowadays we do not have korbanos, nevertheless since Rosh Chodesh from the very beginning possessed this special connection to Kaparah, it is still capable of giving Kaparah for the entire month, as cited by the above Mishnah Berurah.

(3) In summary, we can say that there are 3 levels. (a) Yom Kippur is the highest level of Kaparah because even in the time of the Beis Ha'mikdash the day itself could give Kaparah without sacrifices. (b) Rosh Chodesh is also specifically intended for Kaparah. The chief Kaparah was through the Korbanos in the Beis Ha'mikdash but even nowadays a quality of Kaparah continues (c) Regalim are not chiefly for Kaparah, but rather for rejoicing, but in the time of the Beis Ha'mikdash the chatas of Regalim also gave us Kaparah.

Kol Tuv,

Dovid Bloom

1. I posed your question to one of the Gedolim. He replied that there were some opinions in Halachah who wanted to mention Kaparah in the Tefilah of Yom Tov as well.

2. I do not know who these opinions were, but what I understand from this answer is that there is not such a large essential difference between Rosh Chodesh and Yom Tov as far as Kaparah is concerned, as Kaparah is an important element of both days.

3. In fact, we see in the Torah that the Regalim are a time of Kaparah. See Bamidbar 28:22 and 28:30, where the Torah states that the goat-offerings on Pesach and Shavuos were for atonement. (See Meshech Chochmah to Bamidbar 29:7, who expalins that the reason why the Torah does not mention Kaparah on Sukos is that the pouring of the water in the Simchas Beis ha'Sho'evah is not mentioned explicitly in the Torah, but is a Halachah l'Moshe mi'Sinai.)

4. Therefore, even though the Halachah does not follow the opinions who wanted to mention Kaparah in the Musaf of Regalim, this is because Rosh Chodesh is more clearly connected with Kaparah (as I mentioned in my previous answer). However, the Regalim also have a close connection to Kaparah, even though they do not possess the special quality of Yom Kippur that the very day itself atones without anything else apart from Teshuvah.

Kesivah v'Chasimah Tovah,

Dovid Bloom