The Web Modeler is still in beta and has less functionality compared to the Desktop Modeler.
In the future both might be capable of the same, but for now I'd recommend using the Desktop Modeler.
Are there examples of "real" apps built with the web modeler?
In theory it sounds nice, but I've never really understood why anyone would choose the webmodeler over the desktop modeler.
I've also never seen any proffesional work with the webmodeler.
One main advantage of the webmodeler is that it is a WYSIWYG editor. It's better suitable for editing layout/ui. It's missing the advance editing features of the desktop modeler. You can start using the webmodeler just fine. If you need to do more advanced work you need to switch to the desktop modeler.
Note that at this time the webmodeler is still in beta. I had some syncing issues after enabling the webmodeler for an 'old' model I created in the desktop modeler