[jewbacca]

They're really simple, slight variations yield drastically different behaviour, and some setups are Turing-equivalent (given some liberties of interpretation). This is Wolfram's argument on why they would stand with the Riemann-Zeta function as most beloved to Alan Turing, if he'd only known about them.

More fundamentally underlying his advocacy is Wolfram's obsession with the things, which, 30 years on, has not yielded very much. They don't seem to be terribly useful. His wider investigation of the structure of computation is certainly worthwhile, but CAs, at least in the form Wolfram's written his bible about [1], don't seem to be the philosophical revelation he promoted them as.

[1] http://en.wikipedia.org/wiki/A_New_Kind_of_Science