PWA = Progressive Web App ---"progressive" seems to mean it uses some relatively recent additions to browsers to make the app feel more like a native app.
"Progressive" in this case means it uses progressive enhancement: it'll work even if your browser doesn't have those additions (though QoL won't be as nice), and it also uses responsive design such that it's just as useful on desktop as on mobile.
Viewing mostly on desktop, I hate the new twitter and wish the old one was back. It's a bunch of javascript moving things around instead of a plain old web page (how is twitter's content complex enough that it can't just be static html?), and due probably to some combination of privacy settings i have usually says "There has been an error try again" when i follow a link to a tweet, requiring me to hard-refresh the page (at which point it works).
(In case others didn't know either.)