On a client performance is better. On webmail the response time for every other click in the app depends on your network connection. On a client you can work offline, as well as view + organize mail offline. On a client bulk operations are much faster. On a client you can manage an unlimited number of IMAP + POP accounts in one place. On a client you have more control over security. Notification is also better on a client. If you are the type that lives in your account and needs more features, then clients are generally better. If you only check your account every once in a while then stick with webmail.
On a client calender, notes, to do lists, collaboration tools etc are more integrated and more fully featured. Thunderbird addons make the list of advantages much longer. Gmail works great with IMAP + Thunderbird.