blue gum vs eucalyptus

blue gum and eucalyptus both are nouns.

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

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

blue gum: tall fast-growing timber tree with leaves containing a medicinal oil; young leaves are bluish
eucalyptus: a tree of the genus Eucalyptus

Other hypernyms of blue gum include eucalypt, eucalyptus tree.

blue gum (noun)	eucalyptus (noun)
tall fast-growing timber tree with leaves containing a medicinal oil; young leaves are bluish	a tree of the genus Eucalyptus
	wood of any of various eucalyptus trees valued as timber