Perhaps it was inevitable. Given that ajax and flash enabled the creation of web-based experiences which look and feel like desktop applications, and given that dekstop applications increasingly rely on data and interaction from the Internet, the two worlds are becoming difficult to tell apart.
Is iTunes a desktop application or a web application? It's installed on desktop, so that seems an easy question, but what about the iTunes store? It's really a website, accessed from inside a desktop application. (You could ask similar questions about Songbird, or Miro, or any number of other applications for dealing with Internet media).
Adobe AIR is one platform which can take web aplications, including flash content and ajax-based applications) and connect them to the user desktop. (AIR stands for Adobe Integrated Runtime, making the A in AIR as redundant as the M in ATM Machine).Currently available in beta versions for Windows and Mac OS X, with Linux versions promised shortly after release of 1.0, Adobe AIR applications get a desktop icon, an entry in the Windows Start menu / Mac OS dock, and the ability to raise notifications through system standard approaches, as well as access to local filesystem resources.
Recently, Mozila Labs also took a step in this direction with Mozilla Prism. Prism, available in betas for Windows, Mac OS and Linux, basically packages a lighterweight copy of the web browser, without the usual chrome, which is tied to one specific web application. It also gets a desktop icon, though not (at least not now) any greater access to local filesystem than the browser normally has.
Which is better? As always, it depends on what you're trying to accomplish.
Adobe AIR applications need to be specifically created for AIR. They leverage web technologies (Ajax, flash) but are packaged and deployed as applications. In other words, they're created by developers. It's also got the marketing and deployment muscle of Adobe behind it, so the AIR framework itself (which users install in order to be able to install your application) is likely to find broad distribution quickly.
Mozilla Prism is an end user tool. Admittedly, many of its early adopters are developers, but the model is one in which the user decides to create a Prism application for any frequently used application. The Prism app is just another way of accessing the existing web application. It's got the community good will of the Mozilla foundation and the development strength of the open source community behind it, so it will also likely find broad support quickly, especially among web savvy early adopters of new technology.
In the end, both represent new potential for gaining a foothold on the user's desktop. The more difficult task remains creating an application or content compelling enough for users to want it on their desktop.

