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An about page often describes more information about the person (or organization) that a site represents, or that a site is created by, than on the top level home page.
One of the issues is that, when you are sharing your details with others, you usually do not share the About page. You share the top level page, from which there is usually a link. There may also be some basic information on that top level page which the About page expands on.
For different people/organizations, an about page can indicate different things. For example, some use it to identify interest, contact data, a formalized bio, etc.