There is one advantage of "www" -- if you write the website address without "http://" part, only the "www.appliedrationality.org", some softwares will recognize that it is a web link and will make it clickable; the same thing will not happen with "appliedrationality.org". (For example if you write that in an e-mail, or a comment on a web page.)

I don't know how much this is useful in real life. Could possibly generate a few extra links and higher pagerank.

Either way, some people will write the "www" part even if it's not there, so if it is not the official one, it should redirect.