I'm new here, and I can't find anything about what makes Birthday% different from Any%.

In the previous version of Giftscop (1.1.1) it was not yet possible to progress beyond the birthday scene in the house.

In 1.2, it is possible to reach content beyond this scene, such as the school. To prevent older any% runs from being lost, we renamed the previous any% category to birthday% and created a new one 1.2.

In short, birthday% ends when you unlock the birthday scene in the house, while any% ends when you actually reach the end of the game.