Writing an essay is very different from writing a note, as anyone who ever passed a note to a friend in class could tell you. Of course, if you had passed essays instead of notes, your teacher probably wouldn't have been as upset about it.
The purpose of an essay is to inform others on a topic, and a note sometimes does the same. Notes can have many other purposes, as well. Sometimes a note is a reminder, or a quick recording of a lesson in class, or a personal message to a friend, colleague, or family member. There is no one definition or guide for writing a note, and some notes don't make sense to anyone but the writer. Sentence structure, grammar, and spelling requirements only count for notes intended for others to read, and sometimes, they're not even necessary then. It's not likely, for instance, that your friend will return a note marked in red pen to point out your mistakes.
Essays are structured using an introductory paragraph that includes a thesis statement, three or more body paragraphs with topic sentences that relate back to and explain the writer's purpose, and a conclusion paragraph that sums up the main points of the essay and draws inferences about them.
There is no universal structure for a note. Some notes are as short as a word or two, some are written on paper and some on the back of a hand, and some are intended to be read by others while some will only be seen by the writer.
Essays fall into four basic categories, each of which has its own required structure and guidelines. A persuasive essay argues a point and attempts to convince readers to agree while an expository essay provides facts without any opinion or commentary. A descriptive essay uses words to create a picture in a reader's mind while a narrative essay tells a story.
There are too many types of notes to list, but some common types are notes taken during a class or business meeting, thank you notes, and reminders. While essays can be personal and even entertaining, they are generally more formally written than notes are.
Writing an essay takes research and time, and teachers are usually happy when they see you writing one. Some notes will get you praised in class, but personal ones written during classtime will probably just get you detention.