I have not made this choice and am still using both...
The WebClip has served us well, since it downloads fast which has been helpful during rollouts to get the Mail Profile quickly, without waiting for app downloads.
WebClip has an "Update All" button
The App is searchable.
Probably going to move to the App only next year, since hopefully apps will update auto-magically from the JSS.
The only time I have seen the "Unable to connect to server" error is if/when someone downloads the app directly from the App Store.