Some answer that chodesh is Aramaic, while yerach is Hebrew.
The Malbim and Rabbi Samson Raphael Hirsch (1808-1888) explain that the primary meaning of the word chodesh is not “month” but rather “beginning of the month.” In this way we find that the word chodesh appears in the Bible when one would otherwise expect the phrase Rosh Chodesh to appear (see Num. 28:14, I Sam 20:18, and Isa. 1:13). The word chodesh, therefore, primarily refers to the concept of chadash (“new”) or chiddush (“novel” or “renewal”), and specifically denotes the novelty of the month. That chodesh also refers to the idea of a “month” is only a secondary, borrowed meaning. If Chodesh denotes the beginning of the month, then maybe yerach denotes the entire month as a whole.
The Malbim explains that the word yerach simply denotes a period of thirty days — regardless of whether or not those thirty days represent an astronomical event related to the moon.
Rabbi Aharon Marcus (1843-1916) writes that the word yerach is related to the Hebrew word oraiach (“path”) and to the Aramaic word אתרחיש(“it happened”). He does not explain the thematic link between these words.
Rabbi Hirsch argues that the word chodesh denotes the idea of a month as simply a unit of time (measured by the amount of time it takes the moon’s light to disappear and reappear). In this way, the word chodesh is transcendental, or abstract. On the other hand, he explains, the word yerach connotes the month as a vehicle for maturation.
Interestingly, Rabbi Mordechai Jaffe (1530-1612), the author of the Levush, writes that the custom is to refer to the month on a divorce document (a get) as a yerach, and on a marriage document (a ketubah) as a chodesh.
SOURCE: What's in a Word?, "Old Month versus New Month", see also this page from the Veromemanu Foundation for Biblical Etymology