hajj vs pilgrimage

hajj and pilgrimage both are nouns.

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

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

hajj: 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 hajj include pilgrim's journey, pillar of Islam.

hajj (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