I have often read in FAQ section of websites, that request users to logoff once they are done with their working on website. Have you ever thought why we need that part? Below is the list of thinks I recognize as benefits
User View point:
1) Secure your account, in case your computer is shared or accessed by someone else.
2) Save your personalized settings (well in web world, this statement is not very true, but for desktop application/LAN application it does stand true)
3) Help in Gracefully closing of application.
Developer/Server/Application View point:
1) Allow developer to write a code to free System memory or Resources.
2) On Web allow developer to “kill” the session, so it won’t wait to expire by default timeout, and hence save lot of resources
3) Application gets better life, and usability. Crash little lesser (that I believe no stats available to me though)
4) Web application support more users, as they have more resources.
However, Remember Me feature has little to do with above message, but it do have something to do. Hope to see you incorporate that habit in your websurfing or application usage.