I'm the OP. In the cases where a proposition can be refuted by a proof that doesn't involve counterexamples, by its nature that proof will tell you something about the reason that the proposition is false.

Whereas a counterexample, on its own, proves the proposition false but doesn't necessarily tell you anything else.

The real difference in the constructive case is that there are fewer classes of proposition for which a proof without witnesses is possible.

(Edit: side note, I didn't explicitly say "necessarily" in my original comment. I suppose there could be exceptions, although I'm struggling to think of an example. Constructively speaking, the ball is in your court!)