As libertyedoc has pointed out, many people do use the terms interchangeably. But in regard to George Washington's birthday, many people know that it is February 22. But what is his date of birth? So you see, there are significant differences that transcend your personal use of the terms. Until one can "see" that there is another valid view, it seems ridiculous to assert that it is just as likely. I can "see" your point of view. Can you see mine? http://www.moillusions.com/young-lady-or-old-hag/
It seems that both "date of birth" and "birthday" are accepted but these have slightly different meanings in English. "Date of birth" would usually include the year that you were born whereas a "birthday" is more of an annual event. Is this distinction made in French?
The distinction is stronger in French --- "date of birth" always means "date de naissance" and includes the year, while "birthday" usually doesn't include the year and means "anniversaire", but can in some contexts include the full date, rendering it again like "date de naissance". The French terms are precise and distinct, while the English ones can overlap.
C'mon, know the difference:
Birthday -- happens every year
Birthdate -- the exact date in the past in which you were born. Your birthdays are based on your birthdate.
So, for example, if you were born on January 1, 1985, that is your birthdate. Then the succeeding January 1's are your birthdays.