I don't like webmail, but desktop mail clients have really lagged behind their online counterparts (e.g. Gmail). Desktop mail, in theory, should be faster, more responsive, and more useful when the network is down.
Within the last few weeks, Postbox finally fixed the main bug that was preventing me from using it. (Previously, large image attachments would extend off the screen instead of being dynamically re-sized to fit.) It's pretty great and full-featured, but I'm not really using it yet.
Postbox has some strong competition from Airmail, which is more attractive and is being developed at 50x the pace in Italy. But it's buggy.
I still don't use Mail.app.
I used to conduct my e-mail in Mutt. Those were pretty awesome days. I wish I could go back.