order Primates vs animal order

order Primates and animal order both are nouns.

Word	Noun	Adjective	Verb	Adverb
order Primates	Yes	No	No	No
animal order	Yes	No	No	No

As nouns, animal order is a hypernym of order Primates; that is, animal order is a word with a broader meaning than order Primates:

order Primates: an animal order including lemurs and tarsiers and monkeys and apes and human beings
animal order: the order of animals

order Primates (noun)	animal order (noun)
an animal order including lemurs and tarsiers and monkeys and apes and human beings	the order of animals