It is possible to read email from an email client on more than one computer actually. I have the portable version of Thunderbird, so I can put it on a USB stick and take it wherever I go. Instant access to all my emails.
As for why people use email clients still, well it's a personal preference. Not everyone is totally satisfied with just webmail. I don't see what's so hard to understand about that. There doesn't need to be some huge, spectacular reason to use an email client. It can be as simple as "they want to".
Why I use an email client, I actually go back and forth. I do still use the webmail versions a lot, but using Thunderbird is much more convenient. It allows me a way to access all my email accounts at once (I have 7, 8 if you count my Spartan one, which no longer works at the moment anyway because it's a school email). And without having to log in or out or having to open up a new browser to do so. With Thunderbird, I can be logged in to all 7 emails in one window. Convenient.
I also use it to archive emails to clear up some space, and to clean out some old emails I don't often log into otherwise. It's much easier clearing out an email with Thunderbird as it seems to fit everything onto one page. So it's as simple as highlighting all the emails, right click, and Delete Selected Messages. With Webmail, if I had, oh say, 120 emails I didn't read, I would have to go through multiple pages to delete them all. Oh and I also use it as an RSS feeder; I'm subscribed to some RSS feeds and I check them once in a while. Sometimes there's interesting stuff.