I guess that they would be only marginally better. And be aware that Watson itself is already a cluster of (quite big) computers (citing Wikipedia):
Watson employs a cluster of ninety IBM Power 750 servers, each of which uses a 3.5 GHz POWER7 eight-core processor, with four threads per core. In total, the system has 2,880 POWER7 processor threads and 16 terabytes of RAM
Because of the rules of Jeopardy, Watson was not allowed to use the Web during the game. At some very high level, it contains a digested cache of some of the Web contents. Giving access to the Internet would improve slightly Watson's performance, but not that much.
if you asked the Watson-web to diagnose a person's as-yet-undiagnosed disease, would the web be able to do so more quickly?
Even if by "web" you mean a more powerful cluster accessing the entire Internet, I don't think that it would answer much more quickly & accurately.
AI is a diversity of sub-domains, and is not advancing as dramatically as some believe. It follows a gradual progression, with in some limited fields (playing Go, or Jeopardy) some spectacular progresses. See also Everything but the essential blog entry by J.Pitrat.