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.