I love the Ship of Theseus discussion. And I am definitely on the side of different ship. To me, If you took all those extra parts, and built a second ship and placed them side by side, you couldn't say that they are the same ship. Nor could you take half the pieces from each ship and reassemble them as 2 ships made from old and new parts, and claim they are now the same ship. So if you replaced each plank on the ship, every time you did so, you would have a distinct version of the ship.
Much the same can be used for the people arguement. You arent the same person you were 20 years ago nor the same as a week a week ago. Whether it be physical, mental, or emotional, you will never be the exact same you that existed previously. Sure the differences can be subtle, but they are still different.