<p>I've always been using the cookie-based session store, and never even knew about <a href="http://api.rubyonrails.com/classes/ActionController/Cookies.html" rel="nofollow noreferrer">Cookies</a> until now. So is there any situation where I'd need the cookies hash?</p>

You shouldn't store anything you don't want the user to see or change in cookie. If you store a member ID then the user could easily change the value and pretend to be someone else. Cookies are also sent with every single request to your web server including image, JS and CSS requests. If you are storing lots of information in cookies, this could have an impact on speed.
Cookie-based sessions (in a general context, I can't say I know what Rails does) means your session variables are associated with a session ID which is randomly generated. This ID, and only the ID, is returned to the the user as a cookie. This allows you to associate the users request (because you have session ID cookies)  with the user's sessions. This is safer because it would be very difficult for someone to guess the ID of another user's session.