Kim gave you a good explanation. "Here's today's paper" is less specific and can be used in other contexts. For example, if an office consumes all its copy paper daily, the delivery man bringing the paper for the day could say "Here's today's paper", as to mean "Here is the paper for today."

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.