What makes me the same person today as I was any time in the past? I have new memories and experiences, so why aren't I someone else?

This is, indeed a matter of great controversy, and one that has generated a vast literature. There are those who argue that it simply wrong to say that you are REALLY the same person you were before, because there are so many differences; others who argue that because it is correct to say that you are that there must be SOMETHING about you, or the continuum that constitutes you that undergirds that fact. Others suggest that you identity is a kind of very useful fiction or construct, and so that while it is true that you are the same person now as you were when you were 2, that is true in the same sense that it is true that Ahab was captain of the Pequod, that is because we say so, and this is the kind of thing that can be made true by say-so, as opposed to by discovery. You might enjoy reading, from the Buddhist literature, the QUESTIONS OF KING MELINDA, relating a lovely dialogue between a king and a Buddhist philospher regarding this point, or David Hume's discussion in A TREATISE ...