If I understand correctly, Microsoft Silverlight is a lightweight .NET implementation meant to run on the client side, inside a browser. So now I hear about "out of browser" silverlight applications and I'm confused.
What is the advantage of an "out of browser" silverlight application, compared to a traditional .NET desktop application?
Silverlight 3 out-of-browser apps allow any Silverlight app to have a desktop shortcut, and don't require the browser to be opened to run the app (so technically you could now run the app even if you're offline, since you don't have to fetch the SL app via the web).
With SL 4, you can now have elevated privileges, allowing the SL app to access local resources (such as the network stack and file system).
There's also a simple API call for an out-of-browser SL app to check for updates on startup, and download an update from the server. This could be seen as similar to click-once deployment, but it happens automatically and quickly, so it's more efficient and straightforward than click-once.
Compared to a traditional .NET app (in this case let's compare with WPF, since that's effectively the WinForms replacement), there's very little in the way of installation. No setup program, just the xap file, easily hosted on the web and very quickly installable. SL uses a reduced .NET framework, which might seem like a negative. However, the typical pattern for an SL app is to put most of the heavy-lifting in a service tier. Then, in the service tier you have the full .NET framework and can do pretty much whatever you want (such as accessing databases with ADO.NET).
An out-of-browser install of a Silverlight application still runs in a security sandbox where as a traditional .NET desktop application won't.
The objective of an OOB is to give the user the ability to lift a web-based application out of its browser container and make it easier to access. Its still a web based application. Its worth bearing in mind that this works on a Mac where as a traditional .NET desktop app won't.
This area has become muddier with SL4 OOB that can now ask the user for elevated trust. Now the sandbox is more relaxed and there are greater opportunities to work with the native OS. However there are still many restrictions in place primarily to support multi-platforms. It would not be desirable if it became a defacto that trusted OOBs tend only to work on Windows. It remains to be seen whether that can be avoided.
So if you're thinking of a full-fledge Windows desktop app then you're probably better off using WPF. On the other hand if you don't need full access to the OS, you can deliver via a web page and/or you want to be able run on both Windows and Macs (and possibly other platforms) then perhaps Silverlight 4 OOB+Elevated Trust may be what you need.
I answered a similar question here with Silverlight vs WPF. All of these should apply except for the better html integration.
Libraries are another thing to consider between the two applications. For example, Silverlight 4 natively has built in support for talking to a web-camera and microphone out of the box while WPF and the full .net Framework have a very large third party community of libraries to draw upon which you might need source code for if you wish to rebuild them under Silverlight.
Another factor is limitations in the sandbox, for example, you wouldn't be able to write an application that could connect to any server using any socket in Silverlight 4.