baby vs neonate

baby and neonate both are nouns.

Word	Noun	Adjective	Verb	Adverb
baby	Yes	No	Yes	No
neonate	Yes	No	No	No

As nouns, neonate is a hyponym of baby; that is, neonate is a word with a more specific, narrower meaning than baby:

baby: a very young child (birth to 1 year) who has not yet begun to walk or talk
neonate: a baby from birth to four weeks

baby (noun)	neonate (noun)
a project of personal concern to someone	a baby from birth to four weeks
a very young mammal
a very young child (birth to 1 year) who has not yet begun to walk or talk
the youngest member of a group (not necessarily young)
an immature childish person
an unborn child; a human fetus
a beloved person; used as terms of endearment

baby (verb)	neonate (verb)
treat with excessive indulgence