Disclaimer: I work for Microsoft. However, I don't work on Visio, WPF, CLR or Silverlight team. So, the following is my personal take on these technologies. If you want to quote me, don't do it implying it's the official Microsoft position. :-))
Update: Anything I say below about Flash/Flex/AIR might be wrong, as I have not worked with these technologies and what I know about them is based on what I read on the intertubes. :-) If you notice anything wrong, just shout in the comment and I'll correct it.
To the best of my knowledge, the Visio installer is not built with WPF. It's all unmanaged code; it's just people took a lot of care to make it really polished.
WPF is the new UI platform for building standalone applications for the Windows OS. It supports a declarative UI language - XAML, and related CLR types to program against. WPF is a different platform than WinForms, although it is possible to build applications that mix UI built with both. WPF supports a lot of things that WinForms does not, like bitmap effects, animations, control styling and so on and exposes them both in XAML or through code. Also, WPF relies heavily on vector graphics, as opposed to the pixel graphics in WinForms. In short, WPF is quite powerfull and allows building very snazzy UI. (Don't take my word for it, though, as I am biased; go check around for what people are saying about it or buiding with it. :-))
WPF and WinForms do not compete with Flash/Flex. WPF and WinForms are both UI frameworks for building standalone client applications. As far as I know, Flash/Flex are frameworks for building rich internet applications - RIA (though lately people started interpreting this abbreviation as rich interactive applications).
Adobe did come up with AIR about half a year (or maybe a year) ago, which allows building standalone client applications, so you could say that Adobe is trying to position Flash/Flex/AIR to compete with WPF. Of course, that's my take on it and I doubt Adobe's official positiong is anything like that.
If you want to compare particular MS technnologies with Flash/Flex, take a look at Silverlight - it's the MS RIA platform.
Silverlight is related to WPF in the sense that they share XAML and the corresponding CLR types. Silverlight supports only a subset of what WPF offers, though, as it is not targeting Windows OS only and thus is limited by the fact that it has to be portable.
Quick update to reflect the changes in the year since I've written the answer :-)
With Silverlight 3 shipped, SL and WPF are getting even closer and sharing bigger set of supported features. In addition, most of the new XAML controls are built for platform at the same time. Thus, SL/WPF are getting to a point of singularity...
Also, SL 3 supports out-of-browser applications. In that sense, SL is not only starting to compete with Flash/Flex, but it is also encroaching on AIR's turf.
And no, I still don't work on the WPF or Silverlight team. :-)