The benefit of web based systems is accessibility. You can access it from anywhere and on anything, but you have limits on things like storage and attachment size.
The drawbacks are that, if they choose to, your provider could terminate your service at any time, and because the storage is online and out of your control, you account can be hacked, or subpoenaed, or terminated, and you cannot access your e-mail without a network connection.
The benefits of a client that retrieves your messages from a smtp server is privacy and control. You keep the only copies.
The drawbacks are that most people do not backup properly, so it is easy to loose all your emails, and it requires you to perform other maintenance and oversight, and you cannot access your email if you don't have your own computer with you. It also is likely not free.
Personally, I prefer a hybrid approach. I use a IMAP capable client, (Outlook, Windows Live Mail, Thunderbird) to connect to online web services. This approach means that, should my service ever be terminated, or my network connection fails, I still have a local copy. Should my local copy be destroyed, I can still access it through the web interface from anywhere.