day of the month vs day

day of the month and day both are nouns.

Word	Noun	Adjective	Verb	Adverb
day of the month	Yes	No	No	No
day	Yes	No	No	No

As nouns, day is a hypernym of day of the month; that is, day is a word with a broader meaning than day of the month:

Other hypernyms of day of the month include 24-hour interval, mean solar day, solar day, twenty-four hour period, twenty-four hours.

day of the month (noun)	day (noun)
the specified day of the month	a period of opportunity
	time for Earth to make a complete rotation on its axis
	a day assigned to a particular purpose or observance
	the time after sunrise and before sunset while it is light outside
	a point or period in time
	the recurring hours when you are not sleeping (especially those when you are working)
	an era of existence or influence
	the time for one complete rotation of the earth relative to a particular star, about 4 minutes shorter than a mean solar day
	the period of time taken by a particular planet (e.g. Mars) to make a complete rotation on its axis