We've been using the Network Edition of Zimbra for a few years now where I work. The Network Edition includes support for mobile phones (activesync) and includes a Outlook connector. We use the web client which is pretty rich. And the Desktop Zimbra Client for the laptops which may need access to their email offline.
The Outlook connector is a plugin for Outlook that turns the mapi communication into the native SOAP calls that Zimbra uses. I think they do this so that they only have to develop one communication protocol to the server. Make sense but it's not native Outlook support.