Ashkenazi vs Israelite

Ashkenazi and Israelite both are nouns.

Word	Noun	Adjective	Verb	Adverb
Ashkenazi	Yes	No	No	No
Israelite	Yes	No	No	No

As nouns, Israelite is a hypernym of Ashkenazi; that is, Israelite is a word with a broader meaning than Ashkenazi:

Ashkenazi: a Jew of eastern European or German descent
Israelite: a person belonging to the worldwide group claiming descent from Jacob (or converted to it) and connected by cultural or religious ties

Other hypernyms of Ashkenazi include Hebrew, Jew.

Ashkenazi (noun)	Israelite (noun)
a Jew of eastern European or German descent	a native or inhabitant of the ancient kingdom of Israel
	a person belonging to the worldwide group claiming descent from Jacob (or converted to it) and connected by cultural or religious ties