I'm not a grammarian, but I think that with Rosh Hashana the focus is on the coming year, and it is not a categorization of the day. Rosh Chodesh is a type of day, but not specifically related to the given month.
So Rosh Hashana is "The beginning of THE year", not "New Year's Day", while Rosh Chodesh is a generic "First of the month".
The reason for this is probably because we believe the whole year is determined on Rosh Hashana, but we do not believe that Rosh Chodesh determines the course of the coming month. On the other hand, Rosh Chodesh is special for what it is - the new moon appearing, while Rosh Hashana does not mark any astronomical occurrence, and its significance is only that it marks the beginning of the new year.