first language vs mother tongue

first language and mother tongue both are nouns.

Word	Noun	Adjective	Verb	Adverb
first language	Yes	No	No	No
mother tongue	Yes	No	No	No

As nouns, first language and mother tongue are synonyms defined as:

first language and mother tongue: one's native language; the language learned by children and passed from one generation to the next

Other synonyms of first language include maternal language.

first language (noun)	mother tongue (noun)
one's native language; the language learned by children and passed from one generation to the next	one's native language; the language learned by children and passed from one generation to the next