"For personal one to one text communications I don't see how you can improve on texts/SMS, and for anything else what does twitter do that a web site can't?"
It's the one to many thing -- not "many" as in "countless hoards of fans", but many as in "a set of people I know in real life and who I've run into online" -- most people don't generate enough content to make a website worth coming back to on a daily or more basis, but amalgamated with a bunch of other people's thoughts, and now you've got something!
There are other paths to the same thing -- if everyone used RSS heavily, you could be part of your audience's RSS feed, and still get a proportional amount of timely attention. And Facebook has a similar "fax machine effect" as Twitter -- for close friends, I would hope to get personal email or a call or word in person of important events, but for a big mass of people who I'm not that close to but not entirely distant, FB fills a niche. (That said I barely keep up with FB -- in general it's more "day to day" boring stuff and less people trying to be clever than twitter)
So that's what twitter does that a website (in practice) "can't" - aggregation is the key.
"In fact, I would say it is the communication (real or imagined) with "famous" people that makes it so appealing."
I'm sure this is true for many twitter readers, but it's certainly not universally applicable. I might follow some famous people, but only ones who seem to be trying to write funny or smart stuff.