tooth fairy vs faery

tooth fairy and faery both are nouns.

Word	Noun	Adjective	Verb	Adverb
tooth fairy	Yes	No	No	No
faery	Yes	No	No	No

As nouns, faery is a hypernym of tooth fairy; that is, faery is a word with a broader meaning than tooth fairy:

tooth fairy: a fairy that is said to leave money at night under a child's pillow to compensate for a baby tooth falling out
faery: a small being, human in form, playful and having magical powers

Other hypernyms of tooth fairy include faerie, fairy, fay, sprite.

tooth fairy (noun)	faery (noun)
a fairy that is said to leave money at night under a child's pillow to compensate for a baby tooth falling out	the enchanted realm of fairies
	a small being, human in form, playful and having magical powers