I strongly prefer the native, compiled software of the old days, and dislike webapps. There's only a handful of webapps that I find useful and usable, while the majority feels unresponsive, resource hungry, and generally a step back from late-90s software.
At the same time, I absolutely think it's the online distribution of webapps that is the killer feature. People only need a URL. There's no download and install process (and the less tech-literate users often had difficulties with that), and perhaps even more importantly, there's no patching. As a developer, you know that all your users are running the latest version you got deployed. You don't have version 1.0, 1.01 and 1.12 floating around the Web as downloads, you don't have to beg people to download and install the patches, you just deploy to the production server and that's an automatic update for everyone.
That is IMO why browsers won as an application platform. In my very limited experience coding for the web, it's a remarkably bad application platform. You don't have many useful primitives. You don't really have a choice of language. You're beating protocols meant for documents into submission so they become useful for apps. It's more difficult to create a UI than it was in Visual Basic, Delphi or Borland C++, and so on. But for all of that, you get an application that someone can start using after clicking one link, and that will transparently auto-update.