Just to play devil's advocate here - what the Logout button is doing is clearing a session cookie in your browser. That cookie will eventually expire in any case, but Logout will make that happen faster. At that point, you need to log back in again. That's it. You could clear your browser's cookies to accomplish the same goal - either for all sites or for the public mobile site specifically.
I wouldn't worry about it much UNLESS you are using a shared computer like a library computer. If it's on your own personal phone and you have a passcode set on the phone, it really isn't a significant risk. Log out if you want, but don't feel compelled to do so - again, unless it's a shared computer.
