This can be an important consideration for business purposes, as allowing your email to run through web apps means that although it will be safely backed up, it also means the provider has control over your data, and some companies openly acknowledge that they will scan private emails at least for marketing purposes.
In other words, only you control your emails, and third parties have no access to them unless you purposefully - or accidentally - allow it. While email clients may require a little more work to run, they also allow for more control over user data.
