This is because your Rust program represents the numbers in binary, while the BB(5) champion Turing machine represents them in unary. And unary arithmetic is exponentially slower than place-value arithmetic, which is why we invented the latter. (There are other inefficiencies in the Turing machine, but that's the big conceptual one.)
Nitpicking: just as TMs can use binary or decimal arithmetic, so could Rust programs use unary. It's not an inefficiency in TMs per se, but I can see how it would help a TM to become a BB champion.
> could Rust programs use unary
By not using any math functions except for increment by one?
By representing all numbers with lists (or sets). 0 = [] 1 = [True] 2 = [True, True] Etc. Then for example addition becomes appending two lists together
What else comes to mind in terms of inefficiencies? I can think of quite a few but you seem to have deeper insight as to their ranking so I'm curious as to your thoughts.