GmailUI lets you archive your mail to a folder to keep your inbox uncluttered, you can use a small part of Gmail's keyboard shortcuts and search operators like from: or subject:.
A good reason to use a desktop mail client is to manage more email accounts from a single interface. To make webmail services like Yahoo Mail or Hotmail work with Thunderbird, you'll have to download another extension and follow the instructions from this site. Unlike Yahoo Mail or Hotmail, that don't support POP3*, Gmail does and it's easy to configure in Thunderbird.
Gmail recently added a mail fetcher feature that works only with standard POP accounts.
* Yahoo Mail supports POP3 for the non-US accounts. To get POP3 for the @yahoo.com accounts, you need to pay $19.99 a year for Yahoo Mail Plus.
