One of the best uses of them is simply to make things modal. Consider an example from news.yc. Suppose you aren't logged into the site and you click on an uparrow. What normally happens is you get redirected to the login screen, login and get redirected back and/or actually vote. The whole thing looks something like this in code...
check to see if logged in; if logged in allow vote, else force login with a redirect;
And then you have to start from scratch after the redirect.
With continuations you can do this TRANSPARENTLY like this:
check to see if logged in and if not save my spot in the submit_vote computation, and "push" the login page; login and resume from where the login page was "pushed," thereby voting without redirecting and without restarting the computation.
check to see if logged in; if logged in allow vote, else force login with a redirect;
And then you have to start from scratch after the redirect.
With continuations you can do this TRANSPARENTLY like this:
check to see if logged in and if not save my spot in the submit_vote computation, and "push" the login page; login and resume from where the login page was "pushed," thereby voting without redirecting and without restarting the computation.