AsheGuy, We actually almost completely agree and if it seemed like we weren't before, it was indeed a communication thing. Many moons ago, I wrote a C++ program that communicated with a mail server over POP3/SMTP. I get how it all works under the hood with ports & sockets, etc., however I'm not the best at communicating while trying to target a non-tech audience. Your post is very helpful in clarifying this for others.
There is still a difference I claim in how the client -vs- web email work. While I agree they can offer exactly the same function with how you process/read/delete your email, they differ in the user interaction. The web one is limited to what keys, clicks, shortcuts and touches/swipes the browser supports and the client is limited only by the OS. The web interface can be the most consistent across platforms but the client can be customized for the hardware it runs on.
For a simple example on a PC, right-clicking in my email client brings up context menus for my email and is sensitive to where I do it. On the other hand, right-clicking in my web-based email brings up the browser's generic right-click menu which has nothing to do with my email. From what I've read, this may stem from some browsers not supporting that ability or from developers who don't implement the response in their code.
In both cases, a program communicates with the mail server. The difference is whether the program is the local client or one running on a web server but simply displaying back to the user. I contend that a local client program often provides a richer user interface. Furthermore you can choose from a variety of client options instead of whatever the web email offers.