haj vs pilgrimage

haj and pilgrimage both are nouns.

Word	Noun	Adjective	Verb	Adverb
haj	Yes	No	No	No
pilgrimage	Yes	No	No	No

As nouns, pilgrimage is a hypernym of haj; that is, pilgrimage is a word with a broader meaning than haj:

haj: the fifth pillar of Islam is a pilgrimage to Mecca during the month of Dhu al-Hijja; at least once in a lifetime a Muslim is expected to make a religious journey to Mecca and the Kaaba
pilgrimage: a journey to a sacred place

Other hypernyms of haj include pilgrim's journey, pillar of Islam.

haj (noun)	pilgrimage (noun)
the fifth pillar of Islam is a pilgrimage to Mecca during the month of Dhu al-Hijja; at least once in a lifetime a Muslim is expected to make a religious journey to Mecca and the Kaaba	a journey to a sacred place