In the note Properties of fibers and applications, Osserman writes above Definition 1.5:

Intuitively, the point is that phenomena relating to topology can only change under separable extensions, while phenomena relating to algebra can only change under inseparable extensions.

What is the "yoga" behind this intuition, and where is it explained conceptually?

For instance, why are "geometrically unibranch" local rings 0BPZ "geometric", and why isn't this contradictory to the excerpt of Osserman (for geometrically unibranch we ask the extension of residue fields to be purely inseparable)?