hatchet vs axe

hatchet and axe both are nouns.

Word	Noun	Adjective	Verb	Adverb
hatchet	Yes	No	No	No
axe	Yes	No	Yes	No

As nouns, axe is a hypernym of hatchet; that is, axe is a word with a broader meaning than hatchet:

hatchet: a small ax with a short handle used with one hand (usually to chop wood)
axe: an edge tool with a heavy bladed head mounted across a handle

Other hypernyms of hatchet include ax.

hatchet (noun)	axe (noun)
a small ax with a short handle used with one hand (usually to chop wood)	an edge tool with a heavy bladed head mounted across a handle
weapon consisting of a fighting ax; used by North American Indians

hatchet (verb)	axe (verb)
	terminate
	chop or split with an ax