Each domain name holds seperate cookies, so
www.mydomain.com will hold different cookies than mydomain.com. They are treated as seperate sites, And WWF is setup so that if you login to one, it logs your out of the other. So you will have to log back in each time you switch from
www.mydomain.com to mydomain.com.
As far as redirecting mydomain.com to
www.mydomain.com, it can be done, but you will probably need to contact your web host since they would be the ones who could configure that, typically.