This is not an original idea by any means; it's called the Church-Turing thesis, and has been around for 50+ years.
As for your question "what are those limits?": In a nutshell, no computer program can ever fully answer questions about the properties of other computer programs. Unlike what GP is implying, that is a hard limit we can never ever get rid of. So if weather is universal, we will never be able to fully understand it!
Certainly Wolfram didn't invent the notion of a universal system, nor claim to, nor was that intended to be implied above, as bad as my writing might be. Indeed, on page 1125 of NKS he discusses Church's and Turing's contributions.
What Wolfram repeatedly points out throughout his book is that the threshold for such universality is much lower than one might suspect given the complications involved in, say, a Turing machine. And because that threshold is so low, there exists the possibility that much of the natural world that is not obviously simple is exhibiting universal computation.
As for your question "what are those limits?": In a nutshell, no computer program can ever fully answer questions about the properties of other computer programs. Unlike what GP is implying, that is a hard limit we can never ever get rid of. So if weather is universal, we will never be able to fully understand it!