To make it clear: a birthday is the anniversary of someone's birth: Happy birthday mom! Tomorrow is my birthday. "Birthday" also refers to the date: My birthday is on June 4th. An anniversary is the date of an important past event or the celebration of said event: a wedding anniversary, death anniversary, the 400th anniversary of Shakespeare's death, the 300th Anniversary of New York City. Sometimes, in everyday speech, we can say "birthday" to refer to the day something was created: it's the city's birthday, the nation's birthday, Happy birthday, Telephone!

birthday -- for people anniversary -- for events

Birthday is used for people's birth dates. Whereas, anniversary is generally used more to celebrate a marriage or a relationship between two people. A birthday is a kind of anniversary, but we usually just call it a birthday. Hope this helps