On a desktop app, you can see what kind of request the program makes and which servers it contacts. With a webapp, you'll never know what the app does.
The biggest difference is that it's way easier to see where you're data is being sent in a browser, since it has built in tools. It's very hard to monitor native app traffic that is sent over SSL.
You, Your mail (provider) server.
Vs
You, the server hosting the web app, your mail (provider) server.
Unless you run uMatrix or uBlock Origin (same code in both as far as this feature is concerned), of course.