for me www.mysite.com is much seo than mysite.com this is due to normally all site url will began with www.dot.com, i read about this on article but already forget the link for that article. Correct me if im wrong eg for my site : www.catzer.com/wp - PR3 otherwise : catzer.com/wp - PR2 my site been searching mostly using www. compare to another one.
If your using the www when adding your site to directories and such this would probably be the reason for the higher PR on your www than your straight url without it.
I would use the www version, because that's what the majority of the sites use. If you look at natural linking, people tend to use the www in front of the domain. I noticed this on one of my sites as well. I intended it to be a non-www domain, I had a redirect in place. But still natural links and even offline magazines often added the www to the domain. So I gave up and redirected everything to www by default.
It does not matter which one you use as long as you are consistent and use only one in your link building, although like bisk mentions, many sites will add the www. I prefer the www version. If your domain is long you might want to go non-www. So, redirect one to the other. Choose your preference in Google Webmaster Tools.
Most simple users use WWW, cause they think "Internet is WWW". So if they see something like "www.************" they think it is Internet So I guess it is better to always use the www version, everywhere and all the time
i prefer it without the www. (http://domain.com) and then 301 redirect the www. version to the non www. one.
I like example.com rather than www. example.com because I like using sub domains like videos.example.com think it looks more professional rather than example.com/videos or www. videos.example.com
I don't think it matters which one you use, but make sure to redirect the non www to the www one... EDIT: also, use google webmaster tools to set your preference
if its a new site i will use the www. version of the URL but if i have bought a site that has already been indexed i will check the SERPS to see what has been indexed most with out without the www. But for all my site i allways use 301 to direct it one way or the other, i think that is a good SEO practice.