full moon vs full-of-the-moon

full moon and full-of-the-moon both are nouns.

Word	Noun	Adjective	Verb	Adverb
full moon	Yes	No	No	No
full-of-the-moon	Yes	No	No	No

As nouns, full moon and full-of-the-moon are synonyms defined as:

full moon and full-of-the-moon: the time when the Moon is fully illuminated

Other synonyms of full moon include full, full phase of the moon.

full moon (noun)	full-of-the-moon (noun)
the time when the Moon is fully illuminated	the time when the Moon is fully illuminated