For me it's mainly a speed and convenience thing.
My email client just sits there running all the time (when the computer's turned on) and periodically gathers my email from a number of different email accounts, putting them all together in my inbox for easy viewing. Takes a 2 second click on the task bar to see what's new there and switch back to whatever I was doing if there's nothing of interest.
And fat clients are generally much faster to use than web-based applications, since everything is local. I can flick back and forward between a pile of emails in the same time it would take me to just read a few on the web, especially across multiple accounts. Depends a little on what you're doing though. If you're just reading or typing, then it takes the same time either way, but if you're doing management activities (organising into folders, deleting, etc), or trying to view 4 or 5 emails at the same time to help you compose another, the fat client is much easier and faster.
Of course it's quite possible I'm not using the web-based services in the most optimal way, but I don't care since I prefer the fat client anyway.
Why do you need email stored on your own computer?
Security and convenience.
Webmail providers may be diligent in backing things up for you, but their terms and conditions will basically say somewhere that they can pretty much do whatever they like with your account and your data whenever they want, including such things as deleting your account and all your data without warning, denying you access, and charging you to use the service. Chances are they may never do any of those things (although Hotmail suddenly started charging for POP access years ago), but I want a copy of all my own data somewhere where I know I can always get access to it.
And the convenience thing is much the same as in the previous comment. Searching and manipulating data locally on my own PC is a lot faster and easier than via the web with a browser (for me, anyway).
As for backing up, for me that's relatively easy since I have mostly automated procedures in place for that, and recovery would be quick too. If the hard disk died, I'd just restore an image - a minute or two to start the app and then go and have a cup of tea or dinner while it does its thing (plus a quick unpack of the very latest data, as images are less frequent). A bit more effort if the whole PC died, but then the same would apply to you to get some way of accessing the web again.
Cheers,
GP