I've blogged about GNU screen on many an occasion. I'm not going to go into the technical reasons why I use it, but rather, the personal.
First off, I am in love with the terminal. I am of the opinion that if you want to get into Linux, regardless of your operating system, you better get familiar with the terminal. It's not going away, and no matter how much point-and-click Linux window managers get, the terminal will still be used with tools that you just can't replace or forget about. It will always be there when everything else fails. It's either swim or sink. Your call.
So, with that passion that I have for the terminal, I have a couple programs that I run exclusively in the terminal. Those are irssi, the IRC client of the future, and centericq, one solid multi-protocol IM client. I run both of those programs under the same screen session. I have played with mutt in screen, and even elinks, but I just keep going back to the GUI for my web and email needs. Here's why I keep irssi and centericq in screen:
I want to be immediately available, in case anyone needs me. Whether it be in #ubuntu-utah, #ubuntu-us, any US team channel that I'm mentoring, or on Jabber. If someone needs me, I want to be there, even if I'm not physically at my desktop. Irssi and centericq, coupled with GNU screen allow me to do this. When I'm away, someone can still leave me a message, just as they would with voice mail.
When it comes to web browsing, email, or other applications, I only need the tools when I need them. There is no reason to keep them in screen for me, as they aren't seeking me out. Sure, I keep the occasional vim session in screen when working on a project, but not always. Again, there isn't anything in or from vim that needs me, rather, it me that needs vim.
A couple other great terminal-based clients for IRC would be BitchX, epic and WeeChat (my 2nd favorite IRC client) while gaim-text and bitlbee are great for terminal-based multi-protocol IM (bitlbee actually being an IM client through an IRC gateway using any IRC client- it's just usually paired with irssi).
So, there you have it. I'm perfectly happy with my hybrid marriage between GUI and CLI applications. The idea is online presence and availability. I would highly recommend that you give both of them a try.