Overdue= (1) (of payment) past due; being unpaid by the assigned date or at the scheduled time: The electric bill was two months overdue.An overdue installment.
Be/become overdue is used of payments, library books, video films that you have rented etc.: Even if the books are only one day overdue, you still have to pay a fine.
Expire is used of a license, contract, membership card, etc.: I wanted to pay by visa, but my card has expired.I suddenly realized that my driving license had expired.