The equivalence seems to hold only when the program is complete. Thus, while all proofs can be expressed as programs, not all programs can be expressed as proofs.
I refer to the article on wikipedia which states:
"A converse direction is to use a program to extract a proof, given its correctness. This is only feasible if the programming language the program is written for is very richly typed: the development of such type systems has been partly motivated by the wish to make the Curry-Howard correspondence practically relevant."
Programs are equivalent to proofs given their correctness, rather. Additionally, according to the wikipedia article, extracting a proof from a program requires a very richly typed programming language.
