Now it has turned into an application delivery framework, which uses the old document-rendering abilities to display the UI, with many quirks and workarounds, because DOM was never intended as a performant dynamic medium.
With canvas and WebGL taking more and more, it will turn back into an X terminal, with more advanced network capabilities.
If most important sites switch to canvas rendering, what used to be the browser can be radically simplified and made lightweight and much faster. The 'legacy' full-web sites can be shown using a plugin that does all the fancy and heavyweight HTML 5 and CSS 3 stuff.
Also, with webassembly running JITted platform-independent code delivered over the network, and displaying it in a platform-independent graphical client (née browser), the promise of Java from 30 years ago will finally come true — sans Java proper, though.