Stephen Wolfram, among others, points out that we used to compartmentalize computational problems: sorting a list, taking a square root, and checking whether a phrase is a palindrome looked like three different things. Since Kurt Gödel and Alan Turing we know they are the same kind of problem, instances of universal computation.
One tempting implication is that we can compute anything given the newest models and enough machinery. Wolfram’s computational irreducibility says that is mistaken: for much complex behavior there is no shortcut theory, you cannot predict the outcome, and the program must still be run to get an answer. I wonder how much of that is already implied by complexity theory and by Gödel’s incompleteness theorems and the halting problem, ideas that are almost a century old.
Caring about computation matters because it leads to paradigm shifts. Copernicus moving us away from the center of the universe is the famous example. Before and after him, computing planetary positions with Ptolemy’s epicycles was adequate for most purposes, only less accurate, and more accuracy in our understanding is what a paradigm shift buys.