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.
You can do the same in your browser by opening the developer tools and in fact it comes built in to the browser whereas on a desktop you'd have to install another app like Wireshark/Fiddler to see those requests.
Nothing will. The vulnerability of what happens to your data after it is sent to a third party is the same for any app, web or native.
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.
The logic is a native app can make a direct IMAP connection to the mail server and you can check it is doing so where as a web app has to send the login details to a server which does IMAP for the client and you can not see this last part.