The beginning of the Jewish year is referred to as Rosh Hashanah.
The beginning of the Jewish month is called Rosh Chodesh.
How come Rosh Hashanah has a "hey" (ה) in it, while Rosh Chodesh does not?
I think the answer may be that it goes back to terms used in the Chumash. Rosh Hashana comes from the verse
אֶ֕רֶץ אֲשֶׁר־יְהוָ֥ה אֱלֹהֶ֖יךָ דֹּרֵ֣שׁ אֹתָ֑הּ תָּמִ֗יד עֵינֵ֨י יְהוָ֤ה אֱלֹהֶ֙יךָ֙ בָּ֔הּ מֵֽרֵשִׁית֙ הַשָּׁנָ֔ה וְעַ֖ד אַחֲרִ֥ית שָׁנָֽה
and Rosh Chodesh comes from the verse
הַחֹ֧דֶשׁ הַזֶּ֛ה לָכֶ֖ם רֹ֣אשׁ חֳדָשִׁ֑ים רִאשׁ֥וֹן הוּא֙ לָכֶ֔ם לְחָדְשֵׁ֖י הַשָּׁנָֽה
I think part of the question is your statement "is called". I am cobbling together some thoughts with no real authority so I apologize, but I think it difficult to account for conveniences and popular trends that might not have any halachic weight.
The first thing is that there are contexts in which the noun has no introductory definite article -- the primary one I thought of is the first mishna in Masechet Rosh Hashana. The text reads אַרְבָּעָה רָאשֵׁי שָׁנִים. Since there are four, when discussing any particular one of them, the text specifies which one
בְּאֶחָד בְּנִיסָן רֹאשׁ הַשָּׁנָה לַמְּלָכִים וְלָרְגָלִים
בְּאֶחָד בֶּאֱלוּל רֹאשׁ הַשָּׁנָה לְמַעְשַׂר בְּהֵמָה
So the "ha" is the beginning of a longer phrase which might suggest that the actual name of the holiday we celebrate on 1-2 Tishrei is properly called "רֹאשׁ הַשָּׁנָה לַשָּׁנִים וְלַשְּׁמִיטִּין וְלַיּוֹבְלוֹת לִנְטִיעָה וְלִירָקוֹת" and we just abbreviate it to "Rosh Hashana."
Additionally, the word "chodesh" can be used not just for a month, but for any period of 29-30 days (as we say מבן חדש ומעלה even if the child is not born on a new moon). So is it possible that we are simply saying "the beginning of this counting of 30 days" which would not mean any specific one and would not need a definite article?
Maybe, but it is also important to remember that when we talk about a specific month, we DO use the definite article. The Ibn Ezra, on Hoshea 2:13 writes, "הוא ראש החדש שתתחדש הלבנה". In fact, in our davening, when we commemorate the celebration of the new month, the language is יום ראש החדש הזה (from back in the Yerushalmi, through early siddurim to today). Again, the "ha" is introducing a specific month, not the general idea of 30 days (as in the Abarbenel on Pinchas, "אברבנאל במדבר פרשת פינחס פרק כט ולפי שהיה יום תרועה ראש החדש השביעי").
There is the same 'tension' when referring to the weekly Shabbat. In some cases, people call it "Yom Shabbat"
the Medrash Tanchuma writes, "כיצד ברא הקב"ה ימים וברר לו אחד מהם שנא' (תהלים קלט) ימים יצרו ולו אחד בהם איזה הוא א"ר לוי זה יום שבת"
and there are a bunch of other Aggadic uses of the phrase, whereas other situations, starting in the chumash use Yom HaShabbat.
All of these conventions of naming might simply be evolutions of comfort, not statements of exegetical import.