Outlook Express is used to download mail from a POP3 mailserver. Once the mail is downloaded it's routinely deleted from the server, and you have the only copy.
Webmail (like Gmail) is exactly that - mail that you access via the internet, and read online. Unlike POP3 mail webmail isn't deleted from the server automatically, it stays there until you delete it yourself, or until your allocated space is full, or until it is deleted by your mail provider after a specific time has elapsed - usually a long time.
The advantage of webmail is that you can access it via any internet-connected computer in the world; because it isn't downloaded you leave no trace on the machine you're using.