snow gum vs eucalyptus

snow gum and eucalyptus both are nouns.

Word	Noun	Adjective	Verb	Adverb
snow gum	Yes	No	No	No
eucalyptus	Yes	No	No	No

As nouns, eucalyptus is a hypernym of snow gum; that is, eucalyptus is a word with a broader meaning than snow gum:

snow gum: small to medium-sized tree of Australia and Tasmania having smooth white to light-grey bark shedding in patches or strips
eucalyptus: a tree of the genus Eucalyptus

Other hypernyms of snow gum include eucalypt, eucalyptus tree.

snow gum (noun)	eucalyptus (noun)
small to medium-sized tree of Australia and Tasmania having smooth white to light-grey bark shedding in patches or strips	a tree of the genus Eucalyptus
	wood of any of various eucalyptus trees valued as timber