It is not a case of one or the other. I often implement programs to run on a local server and the GUI is a WebApp. Personally I find QT, GTK wxWidgets to offer minimal advantages over a HTML/JS WebApp. And I can always move from local hosted server to cloud hosted server or back again.
I chose desktop because:<p>I live in the country (limited bandwidth)
I control the source (nobody tells me I violate their 'who knows what')
My information is secure (my desktop app is hosted on non-internet accessible computers)
I have yet to see browser based apps as powerful / native as true desktop apps.<p>Downsides:<p>You have to do more of your own grunt work.
You have to do your own backups.