Using an email client like Evolution make email policy changes a breeze. I use several mail providers and I choose Gmail as default. I want to use a signature below every email I send even as I change which mail provider I want to use to send it. I can do that using Evolution's Preference Window.
You just couldn't do that (easily) with a web mail. You couldn't use OpenPGP to sign and encrypt email without an email client like Evolution or Thunderbird. Not conveniently anyway. If I want to manage my PGP keys I can with a capable email client. If you are concerned about privacy or the integrity of your email communications then you should be using an email client.
I am using Fedora (twenty-two) Linux as operating system and Evolution as my default email client. I do open my web mail through the browser just to check what I'm missing.
Perhaps the elephant in the room when comparing web mail and desktop email clients is speed. Browser scripts have always been faster and crisp. But I prefer desktop email clients when it comes to control over my work and communications.
Use my PGP key for privacy. Privacy is a human right.