If we don’t wear Tefillin on Chol Hamoed, do we wear on Rosh Chodesh? And if so, why is it different?

ב"ה, לע"נ אברהם יוסף בן אלתר משר Shalom, The question you have raised is connected directly to the long-standing dispute of Tefilin on Chol Hamoed. The issue begins with the words of R. Akiva in the Gemara in Menachot 36:. Since Tefilin are called an "OT" a sign, Shabbat and Tom Tov which are also called an "OT" are exempt from Tefilin. This is codified in the Shulchan Aruch Orach Chaim 31:1 as Halacha, that wearing Tefilin on Shabbat and Tom Tov is forbidden because the putting on of the Tefilin which is an OT disrespects the OT of Shabbat and Tom Tov. Based on this, is the dispute between the Rosh and the Rashba. According to the Rosh since Chol Hamoed is not a time in which work is forbidden, putting on of the Tefilin on Chol Hamoed does not diminish or disrespect the "OT" of Yom Tov. But according the Rashba Chol Hamoed is in principle forbidden to do work besides especially defined cases in which you can, therefore Tefilin should not be put on. The different minhagim that ensue between Ashkenazim and Sfardim, Chassidim in Eretz Yisrael and Chutz La'aretz are all well known. Now we get to Rosh Chodesh. The Mishna Berura (או"ח סי' תכ"ג:י) on Dinei Rosh Chodesh explains that just as Yom tov is called an OT, similarly we consider Rosh Chodesh as such but only during Mussaf, therefore the Tefilin must be removed before Musaf. The Aruch Hashulachan (תכ"ג:ה) seems to raise your question and states that he doesn’t understand why we remove the Tefilin on Rosh Chodesh, nonetheless he says that is the custom and one cannot go astray from it. On the other hand, in Hilchot Tefilin, the Aruch Hashulchan mentions the Bet Yoseph (סי' כ"ה) that since in the Kedusha we say "Keter" in the Musaph of Rosh Chodesh which is a crown, it is improper to have on the Tefilin as a crown. The Aruch Hashulchan continues to say that even those who don't say Keter should remove their Tefilin and Heaven forbid to do otherwise because has been established according to Kabbalah. So to sum up, the Tefilin are removed on Rosh Chodesh before Musaf which represents the "OT" , or because we say Keter during Musaph, or because of Kabbalistic reasons. In any case that is the established custom from which we cannot deviate. All the best