What does the "Vary: Cookie" header actually do for serving pages from Varnish? According to the documentation in default.settings.php, and I've seen similar descriptions elsewhere, "Without "Vary:Cookie", authenticated users would also be served the anonymous page from the cache."

However $conf['omit_vary_cookie'] = TRUE; is recommended with Varnish in Drupal 7 and doesn't seem to keep users from being able to switch from anon to authenticated sessions. Is this a header that Varnish doesn't actually care about?
To answer my own question. I've found that with $conf['omit_vary_cookie'] = TRUE;, users who visit a page as anonymous users and log in will see the cached version of the page until they click "refresh" on their browsers. It is a subtle difference that I didn't notice at first as  admininistrator user.