I have to agree that the most important thing we gain from telling our own story is learning about ourselves. Just as we often have to write out a math problem to see all its parts and how they connect to solve it, writing about your own experiences allows you to see them differently and reflect on yourself in a way you may never have before. Relaying our own stories to others brings out how we ourselves feel about those stories and ourselves. Writing is a catharsis for many, especially when you write about yourself.
As for the response of others, I think how important that is, is different for everyone. None of us should be swayed in our opinons of ourselves based on the opinions of others. However, writing is a different way of presenting yourself and may offer an opportunity to get to know someone else by the way they respond to your writing. Sometimes we can connect with someone over what they have written about themselves, when we might not have made that connection otherwise.
Writing about yourself makes the writer vulnerable because it requires facing feelings and emotions in a new way and of course even more vulnerable once it is shared. By as in anything, without risk there can be no reward.