See also: Manage IMAP Folders in Thunderbird
See also: Issues When Using Multiple Clients
Here are a few links that explain the differences between POP and IMAP:
http://www1.umn.edu/adcs/guides/email/imapvspop.html
http://www.guidingtech.com/2207/difference-between-pop-and-imap/
Fundamentally, the differences are:
ISP's like POP because they don't have to store or back up customer mail databases (message stores). Messages are downloaded from the ISP mail servers by the mail clients, deleted from the servers and that's that. The message stores are on the customer PCs and are not the responsibility of the ISP.
As a user, particularly one that works from multiple locations (home, office, jobsites), I feel IMAP is the only choice worth considering.
In the end, I would think one determining factor of which protocol to use is probably the answer to “Do you own your own mail server?”. Though many mail service providers (like us) offer IMAP as well as POP, you may have to pay a bit more for the extra storage and other benefits. If you have your own mail server you probably already have lots of storage and a backup system.
One factor that I see as an impediment to the wider-spread adoption of IMAP is the fact that Microsoft mail clients are generally crappy IMAP clients. I suspect they'd rather sell Exchange Server (messaging server software) for which Microsoft Outlook works very well. I doubt Microsoft really wants you to use IMAP…
My preferred e-mail client is Thunderbird using IMAP. I feel handicapped by anything else. However I'm sure there are many, many other useful IMAP mail clients that some people will prefer. Heck, even the last couple of versions of Outlook (2007, 2010) are much better at IMAP than previous versions. If you just have to have Outlook, you can still use IMAP…you'll just have to learn to like it!
As a final note, webmail is just an IMAP client. Therefore, once you have an organized IMAP message store on your mail server, webmail will display the exact same folders and messages as any other IMAP mail client. That's really nice when you are out and about!