I think that Wolfram is arguing that the study of cellular automata and perhaps similar computational systems could serve as an organizational principle, providing a coherent framework to look at different problem (just like the more familiar frameworks provided by physics and chemistry). This explains the title of his new book, A new kind of Science (i.e. the study of the above-mentioned structures).
On the other hand, Tegmark argues that our Universe is one big mathematical structure. This may be difficult to wrap your head around, but it would mean that we are just mathematical structures that are complex enough to be self-aware and do everything we do. I assume this would not have any observational consequences (as we cannot proof that something cannot be described by mathematics, exactly because we need mathematics to prove anything) and is therefore purely speculative.
As you can see, Wolfram is calling for a new framework to conceptualize and study problems, while Tegmark is positing a theory of the Universe. In my opinion, these are two completely different things. Disclaimer: I have not read the book by Wolfram, nor was I previously familiar with Tegmark's proposal.