To me, a due date and a deadline are sometimes interchangeable but are used in slightly different ways with deadline having more urgency, emphasis and potentially quite bad consequences as a result.
For instance, 'We have to meet that deadline' vs 'the due date for that book return is ____'
Here, we also use due date in describing a pregnancy 'Her due date is September'. You wouldn't interchange deadline in such an example.
Hope that helps.