I once heard a great Dvar Torah that I think is relevant to your question. I'll sum it up for you here and you can look up the sources if you wish.
We see a number of special minhagim related to Rosh Chodesh and Shabbat Rosh Chodesh:
Women refrain from melacha
On Shabbat Rosh Chodesh we add a dish to the seuda (as opposed to Shabbat-Purim where we wait for Sunday to make up for the missed seuda)
On Shabbat Rosh Chodesh we say a special Musaf which includes a section that is very similar to Musaf of a Moed and specifically does not appear in musaf Rosh Chodesh for weekdays:
בָּחַרְתָּ בָּנוּ מִכָּל עָם אָהַבְתָּ אותָנוּ וְרָצִיתָ בָּנוּ וְרומַמְתָּנוּ מִכָּל הַלְּשׁונות וְקִדַּשְׁתָּנוּ בְּמִצְותֶיךָ.
וְקֵרַבְתָּנוּ מַלְכֵּנוּ לַעֲבודָתֶךָ וְשִׁמְךָ הַגָּדול וְהַקָּדושׁ עָלֵינוּ קָרָאתָ
An explanation for the above and some additional questions that I can't remember at the moment could be the Midrash that states that before the sin of the golden calf Rosh Chodesh was a Moed, meaning that there was Isur Melacha.
Women, who did not partake in the sin continue with Isur Melacha.
The seuda on Shabbat Rosh Chodesh is like the Seuda of Shabbat and a
Moed, so we don't postpone the seuda.
When Rosh Chodesh falls on Shabbat and we get a sense of Rosh Chodesh
with Isur Melacha the Musaf changes just a bit to more similar to the musaf of a Moed.