While your grandmother would probably be happy and proud to read your latest essay, she'd more than likely prefer that you just write her a nice letter. The reason is that the audience and point of a letter are usually more personal. If you really want to impress her with how smart you are, you can also send her a letter using the formal structure.
Structure in letter-writing can be quite specific. For a traditional formal letter, an address is printed at the top of the page, a greeting or salutation to the intended recipient or group of recipients begins the writing portion, the body can be made up of single or multiple paragraphs, and a signature by the writer is expected at the end.
The structure of a traditional essay is also specific, but quite different from a letter. Essays begin with a headline, include a thesis statement, have multiple paragraphs with topic sentences that relate back to and further explain the thesis of the essay, and a concluding paragraph that sums up the body's points come at the end.
Essays and formal letters usually have different audiences. Letters are written with a recipient or group of recipients in mind and are not required to make sense to anyone but those recipients. For instance, if you write a letter to your grandmother asking for details on the family reunion, you probably won't have to explain the history of your family reunions. She's been to all of them, so she already knows Aunt Marge will be bringing the potato salad and that no one will want to eat it.
An essay should make a point clear to anyone who reads it. An idea must be presented in the form of a thesis statement and then must be fully explained to an intended audience that a writer assumes has no background information on the subject. That audience can be anyone, there is rarely anything too personal in the communication between a writer and an essay reader.
Like an essay, some letters are written with the intention to inform its readers. However, many letters are written with no other purpose than communication. This is why the tone of a letter is generally far less formal than an essay and rarely requires that the writer provide backup arguments or sources.
Sending a surprise letter to a family member or friend makes you a good communicator. Sending an essay to anyone who didn't ask for one just makes you look like a show-off.