The concept is storing persistent data across page loads for a web visitor. Cookies store it directly on the client. Sessions use a cookie as a key of sorts, to associate with the data that is stored on the server side.
It is preferred to use sessions because the actual values are hidden from the client, and you control when the data expires and becomes invalid. If it was all based on cookies, a user (or hacker) could manipulate their cookie data and then play requests to your site.
Edit: I don't think there is any advantage to using cookies, other than simplicity. Look at it this way... Does the user have any reason to know their ID#? Typically I would say no, the user has no need for this information. Giving out information should be limited on a need to know basis. What if the user changes his cookie to have a different ID, how will your application respond? It's a security risk.
Before sessions were all the rage, I basically had my own implementation. I stored a unique cookie value on the client, and stored my persistent data in the database along with that cookie value. Then on page requests I matched up those values and had my persistent data without letting the client control what that was.