The phrase "a different essay" is correct and usable in written English.
You can use it when referring to an essay that is distinct from another one previously mentioned or considered.
Example: "After reviewing the feedback, I decided to write a different essay that better addresses the prompt."
Alternatives: "another essay" or "an alternative essay".
"I never doubted that Soviet superweapons were vitally important for our country, and for maintaining an equilibrium of forces around the world," he wrote in a different essay.
(This mostly applies to male actors of the Method generation, not to female actors, but that is a different subject for a different essay).
(Why are all these women white? In part, because playing "gross" is a riskier move for women of color a subject for a different essay).
History has effectively erased Wagner's transgression; Mendelssohn is once again almost universally revered, even if many of his works are deemed facile and glib — wrongly, but that's a different essay.
The answer to this question is a topic for a different essay.
Further, in a different essay on terrorism, Walzer rejects arguments on behalf of terrorism by non-state agents that are very similar to those that he accepts as legitimating the bombing of German cities.
I wanted my little backyard to be a welcoming habitat for different kinds of birds and insects in order to be able to share my oasis with these other species (in addition to the resident legions of squirrels and raccoons, which are the subject of a very different essay).
My advice: Write a few different essays about a range of topics, making sure to always connect them back to yourself and how you've grown.
The fact that each college comes up with an individual question - forcing applicants to write as many as a dozen different essays - leaves high school counselors annoyed and seniors anguished.
The bad news: Things that shouldn't be revived might be, which is a little frightening (the reasons "Friday Night Lights" should rest in peace could be the subject of whole different essay).
Novels like 1984 and To Kill a Mockingbird are standards in most high schools, and contain great examples you can use in a number of different essays.
