15th vs fifteenth

15th is not a noun while fifteenth is a noun.

Word	Noun	Adjective	Verb	Adverb
15th	No	Yes	No	No
fifteenth	Yes	Yes	No	No

As adjectives, 15th and fifteenth are synonyms defined as:

15th and fifteenth: coming next after the fourteenth and just before the sixteenth in position

15th (noun)	fifteenth (noun)
	position 15 in a countable series of things

15th (adjective)	fifteenth (adjective)
coming next after the fourteenth and just before the sixteenth in position	coming next after the fourteenth and just before the sixteenth in position