Difference beween Paper & Article

If you are referring to journalism, "paper" would mean a newspaper, and an article would be a subdivision of the contents of the newspaper.

If you are referring to academics, "paper" would mean a required submission from a student, such as an essay. In academics, an "article" would almost always mean something which is offered for publication, or has been published.

If you are referring to government, "paper" would mean a report of an investigative body, and "article" would mean a subsection of a piece of legislation.

To answer questions such as this, we really need context, which is to say, what was the sentence, or what were the sentences, in which you saw these words used?

By Paper and Article and Essay, I mean the context of academics, not the other contexts you referred to. For example, if I want to tell my students to write about a topic, should I say "Please write an essay for next session." or a paper or an article?

I would sugges that in that context, the words paper and essay are interchangeable. However, if you use the term essay, it usually implies a stricture of presenting a logical case for a point of view. Article in such a context usually refers to something which is published or intended for publication, and is almost never properly used to describe a student assignment (i only use the term "almost never" because of the likelihood of someone showing up to dispute it--in the American language, when giving such an assignment, the term article would just not be used).

If you are assigning something which is not necessarily logically rigorous, paper will do fine. If you want the students to logically outline a point of view, i suggest the use of essay.