They pretty much mean the same thing but tend to be used in different contexts. "deadline" is more critical than "due date". If you miss a deadline there can be bad consequences that follow. If you miss a due date, usually there are not bad consequences. Also, "due dates" tend to be used more in school for assignments, whereas deadlines are used in the professional realm, i.e. at work. A boss will give an employee a deadline for a project. If the employee doesn't meet the deadline, then he may get fired. A teacher will give his students a homework assignment with a due date. If the student does not turn the assignment in on time, he will get a bad grade but he will usually not be punished in any other way. Also note that "due date" refers to the day a woman is supposed to give birth to her baby. In this sense, "due date" and deadline are not interchangeable.

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Although they are both technically the same thing, one has more dramatic overtones. Deadline also implies a sense of irreversibility. Once you miss the deadline, there is no other recourse. Deadlines can be used for scheduling the delivery of written items such as news stories, books, web articles, theses, scientific journals, contest entries etc. etc. They can also be used to assign a time or date to explicit demands. Such as a decision on an offered job position, the delivery of a ransom, registering for a contest or a sports event. Due date is used more for scheduled assignments or events that are to be completed or that occur over a specified period of time and delivered at the end of that time. Things like homework assignments, or terms of pregnancy. There is a bit of crossover between due date and deadline. Theses, written articles can use either but the term deadline carries a certain sense of gravitas.

AT SCHOOL Student: When are our reports due? Teacher: The due date is this Friday. Late papers will be graded down for each day that the paper is late. AT THE LIBRARY Reader: Where are my books due? Clerk: Your books are due in three weeks. The due date is December 5. AT WORK Employee: When do we need to finish this project? Boss: The deadline is this Thursday. Our customer will come by Friday morning to pick it up.