bike5,
If the browser supports cookies, the browser can support sessions.
PHP just needs the browser to say "this is my session ID". Cookies work fine for that. Once PHP receives the session ID from the cookie, it knows which session data to load (which was saved to the server's disk (or D😎 on the previous page).
It's also possible (though slightly more difficult) to make sessions work even when cookies are disabled, by putting the session ID in the URL or a form variable. As long as you can get the user to give you their session ID on their next visit (and the session file on the server is not so old that it has been auto-deleted) you can resume the session.
My vote would be for sessions. Much easier to work with. Fewer trust issues too; users can modify cookies, but not session variables (client side versus server side).
Session vars are NOT automaticly gone when the browser closes. Usually the session cookie is set to expire when the browser closes - which means, on the next visit, PHP doesn't know that you have an existing session, and creates a new one - but the cookie can be set to live for a specified length of time. Even when the session cookie is destroyed, the session data still exists on the server, until they get cleaned up automaticly by PHP.