They are both the same. 'Paper' is the shortened form of 'newspaper'. You can use 'paper' when it's understood that you are referring to the newspaper as in, "May I borrow the paper when you are through with it?" or "Here's today's paper" when you have the paper in your hand. However, if you are asking someone for a newspaper, you would say, "Do you have today's newspaper," to clearly distinguish that you are asking about the newspaper and not some other paper. However again, if you are asking about a newspaper at a place that sells newspapers, you could ask for 'today's paper' because again, the reference is understood.