Paper Money Experts: some questions

Is their a difference between a specimen and a proof? What is an essay?

I am not an expert, but…
A specimen is just that: a note that is printed to be a specimen of that particular note. Often for banks in other countries so they know what it looks like.
Essay and proof notes are different types of trial runs to see what the actual finished note will look like.

More info here

A specimen is a reproduction of a banknote that is meant to show what the note looks like, but it is not a note in itself. It can be printed for many purposes. For instance, when issuing a new banknote series, the issuing central bank might distribute specimens to banks, retail shops or the general public so that people can familiarise themselves with the looks of the new notes before they see the real thing for the first time. In many countries, there is a legal requirement to print the word “specimen” or something to that effect when reproducing banknotes (even e.g. in magazines) to avoid potential prosecution for counterfeiting.

A proof is a trial run, a print made from the printing plates to see if the plates are in order. If they are, they are then used to produce the real thing.

An essay comes at an even earlier stage of the banknote production process. It’s an artist’s concept for the potential design of a new banknote, so it differs from the proof in the sense that it has not been made from the printing plates that would subsequently be used for the printing of the real notes.