I guess the title is slightly incorrect. It's not a whole different kind of writing. The words would probably be the same. The mode, however, is completely different.
What am I talking about? Nothing like jumping into the middle of my stream of thought.
I am talking about the Month of Letters, a project started by Mary Robinette Kowal (who does the audio versions of Seanan McGuire's October Daye series, and whose voice I might love a little bit). There's a lot of information about it on the website, but the basic gist is this: every day that the postal service is functioning, wherever you might be, you send one letter or postcard or handwritten something to someone else. You, in turn, receive letters from others. It's a step back to the "good old days" when we weren't all glued to our computer and communicating by SMS and 140-character tweets.
I think it's brilliant.
Back when I was in college, before the internet was quite such a focus of life as it is now, I spent a summer writing letters to a friend. Nevermind that for some reason I had assigned him the wrong last name, I had a blast writing to him and getting letters back.
Since then I've more or less stopped writing by hand. Until recently when I discovered that writing new scenes on the book I'm editing happens much more quickly if I scribble them out on paper. But it really is scribbling, as you may have noticed in my post earlier this month on editing.
So! I think I'm going to join in. I can write a letter or postcard or something a day for 23 days next month. Can you? :)