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.
Possibly only to render the HTML in some emails (and why not, then, just open them in the user's existing browser window,) but otherwise, there seems (to me) to be little about email that requires the overhead of a browser engine and associated Node dependencies and whatnot.
A native app could be smaller, faster, and use less RAM.
What would be the reason to cram the mail client into the browser instead of making it a first-class application on the same level?