Intuitively, if increments become infinitesimally small, why doesn’t Brownian motion become a differentiable function?
Imagine a particle moving around on some trajectory.
Its trajectory being continuous means that as you slow time down, the particle stays closer and closer to where it was: no big jumps.
Its trajectory being differentiable means that as you slow time down, the particle doesn't just stay near where it was, it moves more and more in a straight line.
Differentiability is a much, much stronger condition than mere continuity. As you take a limit in Brownian motion, you get a continuous function -- but you have no guarantees on its direction, which is what you need for differentiability.