child vs baby

child and baby both are nouns.

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

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

child: a human offspring (son or daughter) of any age
baby: a very young child (birth to 1 year) who has not yet begun to walk or talk

Other hyponyms of child include army brat, babe, infant, female offspring, male offspring, man-child, stepchild.

As nouns, child and baby are synonyms defined as:

child (noun)	baby (noun)
a young person of either sex	a project of personal concern to someone
a human offspring (son or daughter) of any age	a very young mammal
an immature childish person	a very young child (birth to 1 year) who has not yet begun to walk or talk
a member of a clan or tribe	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

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