They both mean they have the same birthday.
But they are both ambiguous about whether the speaker intends to say that they are the same age as well. It depends on whether 'birth date' includes the year. The term has no fixed usage that I'm aware of.
You can avoid misunderstandings by saying something like, "They were born on the same date in the same year" or "They have the same birthday, but X is a bit older."