A birthdate is almost the same as a birthday, except the former is always the exact date of birth. Commonly, a birthday refers to the month and day of birth, but it is also common for the same word to refer to the month, day, and year of birth. A birthdate however must include the year along with the month and day. So you may both call April 20 and April 20, 1969 birthdays, but you can only call the latter a birthdate.
If you want, you can still replace all words of "birthday" to "birthdate" while celebrating your friend's birth anniversary through song. It still rhymes perfectly. They might not even notice it, unless you emphasized that "date" too hard.