Paper is more like a slang word that is widely used when talking about an essay. It usually means the same thing
EG: "The paper is due in a week" and "The essay is due in a week" mean the same thing in this situation.
(You may already know this, I just like to be thorough) However, paper is also an object and doesn't always mean 'essay'.
EG: "The child drew on the paper"