How is this any different from every other desktop mail client (eg. Thunderbird, Inky, Evolution, Notmuch, Sup, Mutt, ...)? What makes this better?
Contrasting Mailpile with other tools, one difference is that the basic design is that of a search engine, not a tool for reading mail from folders. Most current desktop mail clients are built on top of a bad paradigm, in my opinion.
Another exciting thing about this model, is that since the UI is a website (of sorts), we can leverage the collective experience and creativity of the web design community. That is a much, much larger pool of talent than UI designers who know C++ or Objective C, or whatever.
Finally, making the app a web server means you get an API to interact with your e-mail almost for free.