A birthday is an anniversary of the birth of a person or other entity in our midst. Celebrating one's birth is wide spread and crosses many cultural boundaries and has many inherent traditions depending where you are.
Often there is a gift or party or some rite of passage which transpires during the event. Religions are replete with celebrations of birthdays — such as that of Christ, Buddha, and others notable to the founding of the various religious sects.
While the birthday is a recurring occasion, the birthdate is totally singular. Contrast the recurring celebration such as August 23 being my sister's birthday while August 23, 1956 was her birth date.
The birthdays recur while the birth date remains static.