Google Wave is a fascinating communication service, and Waver is an Adobe Air based desktop client. It's cross-platform, as are all Air applications, and allows you to access the service without a browser.
The advantage of using wave outside a browser are mainly aesthetic - it may aid concentration! Waver doesn't add to the service at all. You have two tabs - one for contacts and another for 'waves', which is Google's term for projects/conversations in Wave.
While Waver is functional, it feels cramped compared to the browser experience, and less intuitive. Also, it doesn't solve Wave's notification issue. You still need to be watching to know if a wave has been updated!
Another issue with Waver is that even though it's only in version 1.0, it does look and feel unfinished. For an Air based application it's really not pretty. Perhaps as more people use Wave and it becomes clear how they want to use it, Waver will develop more, but it's hard to see the point of a desktop client that's basically worse than the web app.
Waver puts Wave on your desktop, but at the moment it's hard to see why you'd want to leave the in-browser version.