A constructive proof of existence is exactly a program. (It might not always be a program for a Turing machine, because "constructive" and "computable" are not exactly the same - but that's beside the point.) Even a non-constructive proof of existence for x can be significant in a programmatic context; it tells you that you can posit an oracle for x (e.g. asking for it to be input by the user, introducing further assumptions based on some special case, etc.) without thereby crashing the program or causing it to behave incorrectly.