Richard Monson-Haefel in a recent article "AJAX RIA World - The Tale of Two Webs" challenges the assumption that all web applications must be browser based to be considered real web applications. Indeed with advent of more sophisticated RIA platforms like Adobe's AIR and Curl's Nitro the line between client server and web applications is blurring.
Both AIR and Nito applications can be delivered over the web, installed on the desktop, executed outside the browser and run off-line. These application behave like desktop applications but have the TCO attributes of web applications. They have the best attributes of both client-server and web applications. At Curl we've been referring to this new class of applications as "Fit Clients." They're not FAT or THIN but FIT. We've been using the following graphic to explain the convergence of applications types into this new class.

While both client-server and browser-based thin clients have been with us for some time, Web2.0 technologies like Ajax and desktop widgets (client-installed and web-based) are creating the realization that it's no longer either or. I think within a year we'll start to see broad deployment of these Fit client applications as more and more enterprise users will expect the benefits of both and the world of two webs will start to look like one again.
This trend will clearly disadvantage Ajax based applications as they lack the key features of Fit clients including desktop installation and off-line execution. Already many (see here and here) are recognizing the limitations of Ajax for implementing enterprise applications and I expect this realization to continue to erode Ajax's use in the enterprise.