On the desktop, I will generally opt for using a web site over an application where possible, except when an application is seriously more convenient. For example, at home I typically rely on webmail and forgo using a local client; at work the inverse is true. GMail is one of the best ways to do e-mail :-).
When it comes to e.g. Android, I will often opt in favour of a decent app, over the web version. Apps generally have less overhead then rendering a desktop-oriented website, heavy on JS and meant for the mouse, yet mobile apps sometimes suck compared to websites; mobile or web version. When it comes to Google stuff, I use the apps. The mobile versions (at least on my TF101) are fairly similar to the mobile apps. The desktop *<b>web</b>* versions are best, but the Android apps are O.K. Contrast: the Facebook app has always been crap, compared to either version of the website.
The advantage of going web with it, is that the cloud auto-magically can solve a lot of little logistical problems. Like keeping data in reasonable sync, favouring search over storage, and removing the issue of "Maintance" -- all you need to keep up to date is usually your web browser, once in a blue moon. It is even possible to have web applications that are aware of one another, although this is sorely under-utilized in reality, versus what can be done. Facebook stuff is probably at the forefront, in more ways then one (but not all of them awesome).
When it comes to using an App, you gain hopefully better Native Integration. On the desktop, not so much in my experience: stuff like Mozilla Prism and Chromes app mode just don't cut it just yet, nor does the whole add-on/extension crapola. I'm talking a *<b>Real</b>* native application. Thunderbird versus GMail. Thunderbird will always have better integration with today's PC environments than the GMail's web app. Well, knowing Moldy old Mozilla maybe not always, but you get the point ^_^. The few exceptions that I've seen are products like Dropbox, where the native application _is_ the major selling point of the experience, and thus makes the cloud go round. It just happens to have a useful web user interface to boot.
Native applications open up a whole world for interesting savings and mangles. A good example, using an Android phone, you can have device local, mobile carrier, MS Exchange, Google, and Facebook contacts all synced even though they come from seperate providers; anyone could extend their service like this, fundimentally. That's how stuff should be done. In some cases (Google, Exchange) it is even possible to have calendar data synced. The downside is that doesn't get merged back into the Google-cloud, and that is probably a Good Thing(tm) even if I and others might like the option of it.
I think the web is one of the better interfaces for creating applications, if you want a UI that isn't better mapped to command line programs. Android offers quite a nice programming model, and I assume that iOS and BlackBerry-land offer something sufficent. The world of PCs not as good.
Personally, I think in the future, we will see the evolving "Mobile" experience rise up and destroy both the PC and Web 2.0. The difference is, it will be Mobile 3.0 :-). Then this will all fade away until such a millienia that notions like PC, Mac; Desktop, Laptop, Phone; all of it fades away into being about as interesting as using cornkobs to whipe your ass instead of the three sea shells.
Given the choice, it's obvious where I stand. But the real question is, does it run unix? :-P