The two approaches are quite different under the hood.
After using both for a few days I prefer Fogger (at least at this stage of development).
Ubuntu Web Apps
Ubuntu Web Apps uses your preferred browser (Firefox or Chromium) to view sites utilising a browser plugin to create a page-specific shortcut in the unity launcher and to access unity desktop features (notifications, messaging menu, sound menu, dash, etc.).
- One advantage of this approach is deep integration with the desktop.
- One disadvantage (IMHO) is that web apps aren't given their own separate browser instance, but are just opened as a new page in Firefox or Chromium.
This means, e.g., that when cycling through open apps with Alt-Tab your web app simply appears as another browser window.
It also means that in some situations the web app seems to open as a new tab in the browser instead of creating a separate browser window.
All of the usual browser 'Chrome' (statusbar, URL bar, bookmarks bar) also appears in the app's browser window, meaning that you don't feel like you're using a desktop app--you're clearly using your browser.
Finally, you can only "integrate" a web page if your browser gives you the option, i.e. if someone has set up an integration script already.
Fogger creates its own new browser window, with a new application instance for each web app and uses two kinds of files to customize the web page's behaviour:
The user styles basically override the css of the page, customizing the page's appearance (see the Fogapp for Facebook).
A big advantage of Fogger is that you can turn any page into a Fogapp on the fly, in a matter of seconds. For those who like visual tweaking, it's also easier to use a custom icon for a Fogapp, and Fogapps respect your icon theme where it has an applicable icon (e.g., for gmail).
In the "I don't know which is better" category, the two frameworks take very different approaches to your user data.
Ubuntu Web Apps try to centralize data. Not only does your web app use the same data profile as your default browser, it also centralizes account data at the OS level via Gnome's "Online Accounts" manager.
Fogger, on the other hand, isolates each app's data. This makes it easier to, e.g., keep your browser signed in to Google but use a Fogapp to surf YouTube without being signed in (to avoid Google's nasty data collection).
It also makes it easier to set up multiple web-apps for the same site, each of which uses a different user profile. This could be especially useful if multiple users employ the same computer and want separate apps for "Mary's Gmail" and "Bob's Gmail". It also makes it easy to clear the data for one web app while leaving your other data untouched.
So at present, I think Fogger provides a more flexible and satisfying kind of integration, even though at present Unity's Web Apps offer deeper access to the unity desktop.
There's nothing stopping Fogger from expanding its API for user scripts, though, and I hope that happens. Actually, it would be great to see the two approaches be united (unity?), with the current differences becoming different options you could choose when creating an app. This is one instance where I'm not sure competition is necessary.