In the 'paper' would be used only when describing a piece of academic writing that you submit to a journal. It is not interchangeable with the more generic term 'essay' here.

It depends on the country. In the US these terms can be interchangeable- they are both pieces of writing. In New Zealand, an essay is a piece of writing you submit and a paper is a University course that you take. I am not sure about the UK.

An essay is usually written in response to a question or proposition. The essays are generally written in an educational and analytical tone. . It discusses both the arguments in favor and against of the topic. The essays usually do not need any formal format and no headings and sub-headings are used. The purpose of an paper is to inform the readers of a concept. The paper are written in a conversational tone so as to inform the readers of a concept in an interesting manner. The paper however usually follow a heading and sub-headings format to make readers understand the concept easily. Reference :- http://gk.queryhome.com/7167/what-is-the-difference-between-paper-journal-and-easy?show=7174#a7174

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"

（1）paper作为可数名词意为论文的时候，它一般是指在学术会议上宣读的单篇论文（区别于发表在期刊上的单篇文章article)。essay的意思是短文、随笔，它的题材不限，当然如果是关于学术、科技的内容，也可能就是一篇小论文，但是形式不想标准论文格式那样严格，语言也可能比较随意。 （2）有的课程按规定结业的标准是写一篇读书报告（当然要活的任课老师批准通过），有的是与结业考试结合，各占一定的权重。这种作业可以叫paper，也可以叫essay, 所不同的只是paper的学术意味少强一点，最好搞得像一片论文似的。essay则可随意一点，当然这是字面意思，具体的要求由课程大纲和老师规定。