Any% means you get all the requirements from Dumbledore to pass the year and reach the end credits. This includes all Challenge Shields, all Portrait passwords (so you need a certain number of Cauldron Cakes), All Pixies defeated from the well, and all Buckbeak sessions (I think there's something else but I don't remember what it is).
Better Any% is more a low% category that doesn't reach the credits; you get to the end of the story (Dumbledore checking the requirements) but you don't reach the end credits because you don't pass all the requirements. It's been called Better Any% because some runners dislike having to get all the requirements for Any%.
Though it would make sense to call it Low%, we still do not technically beat the game persay. We just get to McGonagall and Dumbledore as soon as possible. So in a way, it is a "Better Any%" category
