blue gum vs eucalypt

blue gum and eucalypt both are nouns.

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

As nouns, eucalypt is a hypernym of blue gum; that is, eucalypt 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
eucalypt: a tree of the genus Eucalyptus

Other hypernyms of blue gum include eucalyptus, eucalyptus tree.

blue gum (noun)	eucalypt (noun)
tall fast-growing timber tree with leaves containing a medicinal oil; young leaves are bluish	a tree of the genus Eucalyptus