Islam vs Ash'arism

Islam and Ash'arism both are nouns.

Word	Noun	Adjective	Verb	Adverb
Islam	Yes	No	No	No
Ash'arism	Yes	No	No	No

As nouns, Ash'arism is a hyponym of Islam; that is, Ash'arism is a word with a more specific, narrower meaning than Islam:

Islam: the monotheistic religious system of Muslims founded in Arabia in the 7th century and based on the teachings of Muhammad as laid down in the Koran
Ash'arism: the classical synthesis of Islamic philosophical theology, formulated by al-Ash'ari.

Other hyponyms of Islam include Mahdism, Salafi movement, Salafism, Shiism, Wahabism, Wahhabism, Alawism.

Islam (noun)	Ash'arism (noun)
the monotheistic religious system of Muslims founded in Arabia in the 7th century and based on the teachings of Muhammad as laid down in the Koran	the classical synthesis of Islamic philosophical theology, formulated by al-Ash'ari.
the civilization of Muslims collectively which is governed by the Muslim religion