The iPhone uses IMAP for services like Yahoo. Way better than a web interface on the phone. On the desktop, I really notice no problems with IMAP. It takes a few seconds to get most messages, then access to them instantaneous unlike via a web interface.
My offline access is mostly when I'm travelling, trains, planes, or overseas between paying for internet. Even commuting to work has a significant length of time underground on the Tube with no internet access, but that's where I do most of the emailing from my phone. Longer emails are typically to distant friends and family. More complex emails might just be sending a bulleted list to my wife (annoying to do on a mobile device), or even a copy and paste of a table from our online banking account which works better in a MUA than web interface. Sounds pretty Joe Average, although a lot of people these days don't seem to realise how shit the user experience is away from a MUA because that's what they've mostly experienced.