Web based mail (webmail) is no particular system, but depense on the mail service of your host. Several good systems are around, and many of them is almost similar to MS Outlook, not least to mention Microsofts own webmail solution, called Outlook Web Access (OWA).
However you should distinguish between mail system like this instead, because Outlook can be configured in many other ways than what I reckon you must mean distinguishes outlook.
There is the kind of systems that stores your messages, making them available on-line and everywhere. Such standards are named stuff like IMAP, MS EXCHANGE.
Others are computer stored systems that will download the message from the server for viewing, making them only available on one computer once downloaded. This standard is usually SMTP...
Thing is that MS Outlook can be used as both IMAP, EXCAHNGE and SMTP! So if you like outlook interface, then it depends on the services from your host... Often you can access your account as both IMAP and SMTP.
So there are different storage methods, and there are different interfaces. Web based mail is a storage method, and Outlook is an interface.
I personally like MS Exchange that centralisez all outlook tasks. So you can share all info in outlook, and also between users. However this can be a pretty costly affair. Then go for IMAP instead with a webmail interface. You can then use Outlook, and webmail to access the same account.
Dont use SMTP unless you have a good reason for it. It is outdated, and it is really annoying that your mail wont be online once you have connected...