adjective
BAD : I suddenly realized that my driving licence was overdue.
GOOD : I suddenly realized that my driving licence had expired.
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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 licence, contract, membership card, etc: 'I wanted to pay by Visa but my card had expired.'