An essay is essentially a collection of paragraphs, and as has already been said, a pargraph is essentially a mini-essay.

An essay contains an introduction, a body, and a conclusion.

A paragraph contains an introductory sentence, a body, and a closing sentence.

An essay is a collection of thoughts/ideas on a topic.

A paragraph is one thought/idea.

In many ways, a paragraph is a mini-essay. Both have a limited subject with (a) precise opinion(s) In writing either a paragraph or an essay writers do three things:

1. Writers tell the readers what  they are going to tell them
2. Writers tell this to them and illustrate/prove it, giving details to explain or develop the support
3. Writers then tell readers what they have just told them.

• Both a paragraph and an essay make a general statement that contains an opinion (the essay may have three). 
• Both paragraphs and essays must be unified (all ideas relate to the general statement)
• Both paragraphs and essays must have coherence (there is explanation of the support, reminders of the opinion, and transitions). 
• Supporting sentences give reasons, examples, statistics, testimony, or facts that prove the veracity of the opinion stated in the general statment. 

• A conclusion rewords the general statement (and topic sentences of the body in an essay) and its opinion(s) with a "clincher," a thought-provoking sentence that relates to the general statement and extends its idea, or reflects upon its idea.

As a writing teacher with an aversion to the extremely structured paragraph (though it is absolutely valid and effective, just sometimes dangerous in the wrong hands!), I would say that they are similar because both of them are trying to get across a certain idea.  An effective essay can be short or long or have three paragraphs or nineteen.  The point is whether it accomplishes its purpose or conveys its idea effectively.

So too, a paragraph can be twenty one sentences or it can be one word.  The point is that both of them should have a purpose and should be trying to get across a meaning.  If you aren't, then what are you wasting our time for anyway?

In my opinion, the similarity between an essay and a paragraph is in the way that the two things are (or should be) structured.  I believe that an essay and a paragraph should be set up in just the same way.

Both an essay and a paragraph must start with an introduction that tells what that is about.  In a paragraph, it's just a sentence or two.  In an essay, it's usually a paragraph.

Then both kinds of writing need a body.  This is made up of the sentences or paragraphs that do what the introduction says is the purpose of the piece.

Finally, both should end with a conclusion.  This is very similar to the introduction -- it just sort of summarizes the major argument of the piece of writing.

What are the similarities and deferences that exist in a paragraph and an essay

Besides the obvious difference of length, there are fewer disparities in the concept of writing a paragraph and an essay and more similarities.  In fact, there is a form that Edward P. Bailey and Philip A. Powell in The Practical Writer term a "One Paragraph Essay."  With such a paragraph, the writer creates a topic sentence that in its general statement contains an opinion.  This opinion, then, is supported with any of the following; a quick example, a narrative example, a statement by an authority, a statistic. As a conclusion, the writer rewords the topic sentence with what is called a clincher, a thought-provoking remark. Here is the outline for such a paragraph:

• Topic sentence
• Specific support
• Specific support
• Specific support
• Reworded topic sentence with clincher 

Thus, the concept of a paragraph is to state to the readers what you are going to tell them, then tell it to them, and then, reiterate what you have just told them.

Essentially, an essay has the same purpose, just in greater depth and extent.  However, in a five paragraph essay, for instance, there is an introductory paragraph which is not included in the writing of a single paragraph.  In the introduction, the thesis is similar to the topic sentence, but it contains 3 opinions on the general statement, rather than just 1. 

Of course, the three body paragraphs are much the same as the one-paragraph except that they do not have a reworded topic sentence as their last sentence as they are mainly the supporting paragraphs.  Instead, the last sentence is usually a transition sentence that includes points from the paragraph that can lead to the next paragraph.  Thus, the writer connects an idea from the written paragraph that will launch the next paragraph.

Finally, the last paragraph is the concluding paragraph.  So, instead of one sentence that rewords the topic sentence as in a paragraph, the writer ties all the main ideas of the body paragraph together and then rewords the thesis.  Again, a clincher ends the essay just as it does the paragraph.

Here is an outline of a five-paragraph essay:

• Introduction with motivator and thesis statement containing 3 opinions
• First Central Paragraph with topic sentence, and specific supporting sentences (4-5) and transitional sentences. Transitional sentence to the next paragraph
• Second Central Paragraph [same pattern as the first]
• Third Central Paragraph [same pattern]
• Conclusion with reworded thesis statement and clincher

So, the paragraph, in essence is a mini-essay.  They are similiar in that they have the same purpose of communicating an idea with support.  Here are the differences: 

1. Whereas a paragraph has a single topic sentence with a single opinion, an essay has a thesis statement that contains three opinions. 
2. Whereas a paragraph has a support as a single sentence, in an essay, a support is an entire paragraph. 
3. And, whereas a paragraph has a single reworded topic sentence as its concluding sentence, the essay has a paragraph.

Source:  Bailey, Edward P. and Philip A. Powell, eds. The Practical Writer.  Boston:  Thomson Higher Education, 2008.