Yes, they are different. A paragraph can be a complete piece of writing only a few sentences long. Or a paragraph can be part of a longer piece of writing, in which case it might be only one sentence long.
But an essay is a longer piece of writing. It is usually defined as a short piece of writing on a particular subject. Students are often taught to write five-paragraph essays. This may be a good writing exercise, but in real life an essay is rarely limited to five paragraphs.
In the discussion that follows, I will compare a paragraph that is a complete piece of writing to a five-paragraph essay that students are often asked to write. They have a similar structure:
In a paragraph, the introduction is usually the first sentence, called the topic sentence because it tells us what the paragraph is about.
In an essay, the introduction is a paragraph. It tells us what the essay is about, but it might also define something, or give an interesting story or example.
In a paragraph, the development is limited to one or two sentences for each idea. This means that there is not a lot of room for details.
In an essay, each idea is developed in a separate paragraph. Usually the first sentence of each paragraph states the idea to be developed. The paragraphs often start with connecting words such as Firstly, Secondly, Lastly (or in American English, First, Second, Last). The next sentences within a paragraph give more details.
In a paragraph the conclusion is usually one sentence. It might be a summary, a closing remark, an opinion, or a question to the reader. It may be very similar to the opening sentence.
In an essay, the conclusion is a paragraph. Like the conclusion of a paragraph, it can be a summary, a closing remark, an opinion, or a question. The introduction and conclusion are often very similar.
This structure is often taught in school even though most real writing does not follow this structure rigidly. The reading passages in the textbooks are examples of short essays, but some of them follow the above structure more closely. Read some of them and see how the essays are constructed and how the different paragraphs relate to each other and to the whole. The following passages in the textbooks follow the above structure closely:
Sec 1, Unit 6, page 33, “Cars without oil”
Sec 1, Unit 16, page 103, “Who needs satellites?”
Sec 2, Unit 4, page 23, “Cooperation”
Sec 2, Review F, page 118, “Are dinosaurs coming back?”
Sec 3, Unit 9, page 53, “Unusual weather”
Sec 3, Review E, page 98, “Marie Curie was my here”
See also here
about writing paragraphs and here
about writing letters.
To find more information on the Internet, type how to write an essay, or five-paragraph essay, or similar phrases into a search engine.