The Ship of Theseus

The famous ship of the hero Theseus is preserved in a museum. Over the centuries, its old wooden planks begin to rot. To preserve it, each rotting plank is carefully replaced with a new one, identical in every way. After 100 years, every single plank on the ship has been replaced. Is the ship in the museum still the Ship of Theseus? As a follow-up, a collector has gathered all the old, rotten planks and reassembled them. Which of the two ships is the real Ship of Theseus?