We recently bought half a salted codfish. I did a Spanish recipe (and asked a question about it). The fish had a very nice texture (although it was way too salty).
This weekend, my wife did the same Spanish recipe, with the same codfish... but the texture of the flesh was chewy (her fish was less salty).
Any idea why? My guess is that the fish was overcooked. But I'm not sure.
Assuming the cod was the same quality as before and you de-salted the fish in the fridge for 24 to 48 hours, then I would also guess overcooking. Rendering the gelatin at 45°C (as suggested in the paper @yossarian uncovered) should prevent the fish from cooking and give enough time to get it into the oil. You can always cook the fish in a separate step after you made the pil pil.
Modern techniques for salting cod use less salt and create a product that will not be edible by future archeologists, so store in a dry cool place (tightly wrapped in the fridge).
A complete guess here, but I'm assuming the process of curing the fish changes the protein structures. The salt denatures the proteins which drives out the liquid in the meat to cure it. That process will result in tightly coiled protein molecules which will have a chewy texture similar to beef jerky.
