I am very inexperienced in this particular part of formal set theory, but I was always informally told that the main reason the distinction between constructive and non-constructive proofs is related to the Curry-Howard Correspondence, which informally states that constructive proofs can be rewritten into computer algorithms, whereas non-constructive proofs can not.
https://en.wikipedia.org/wiki/Constructive_proof
I am very inexperienced in this particular part of formal set theory, but I was always informally told that the main reason the distinction between constructive and non-constructive proofs is related to the Curry-Howard Correspondence, which informally states that constructive proofs can be rewritten into computer algorithms, whereas non-constructive proofs can not.