To "make a deadline" means to finish something by or before it is due.
People also use "meet a deadline" in the same way. When you use "make" it sounds like it was more difficult to finish on time. So you'd probably be more likely to see "make" in a sentence like this:
I just barely made the deadline.
And you'd see "meet" in a sentence like this:
Aaron's usually pretty good about meeting his deadlines.