Quora is usually boring but there are some interesting opinions here.
Future of Internet: What will the Internet look like in 2020? - Quora
Future of Internet: What will the Internet look like in 2020? - Quora
My guess? The Internet will be omnipresent; the web will be 'dead'. The web as we know it is built upon, and limited by, the aging paradigm of static pages. It hasn't yet reached its limit, but as we approach newer ways of interacting with the Internet, the web becomes more and more constrained.
The Internet will probably become more of a true protocol rather than a term used interchangeably with 'the web'. Browsing the web is already more accurate than saying 'browsing the Internet', and that's a process that'll continue. Everything can and should be connected to the Internet, with almost no exceptions. Should it be public, though? Not necessarily, but that's easy - just keep it on a secure connection and off the web.
One only has to look at apps (more and more people use Internet without accessing the open web) or touch interaction (a finger and a mouse pointer requires different approaches in UI) to see that this is already underway.