birthday gift vs birthday present

birthday gift and birthday present both are nouns.

Word	Noun	Adjective	Verb	Adverb
birthday gift	Yes	No	No	No
birthday present	Yes	No	No	No

As nouns, birthday gift and birthday present are synonyms defined as:

birthday gift and birthday present: a present given in celebration of a person's birthday

birthday gift (noun)	birthday present (noun)
a present given in celebration of a person's birthday	a present given in celebration of a person's birthday