paediatrician vs baby doctor

paediatrician and baby doctor both are nouns.

Word	Noun	Adjective	Verb	Adverb
paediatrician	Yes	No	No	No
baby doctor	Yes	No	No	No

As nouns, paediatrician and baby doctor are synonyms defined as:

Other synonyms of paediatrician include pediatrician, pediatrist.

paediatrician (noun)	baby doctor (noun)
a specialist in the care of babies	a specialist in the care of babies