I've just completed my INSEAD essays and realize that I've taken a somewhat casual approach when telling my stories.
Do you think this is fine or should the essays have a more formal tone to them?
Hard to say without seeing what you mean frankly, but I'd venture away from casual. Try and avoid writing your essays like you would speak in a conversation - it makes them wordy and superflous. At the same time, you don't want them to read like a legal brief, so some friendliness is OK.
I'd err on the side of caution here and write more formally - you are pitching yourself (a business sale) not talking to a buddy at a bar.