Why isn’t Brownian motion differentiable? Intuitively, if increments become infinitesimally small, why doesn’t Brownian motion become a differentiable function?
 A: 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.
A: Here's another intuitive explanation using self-similarity.
We know that $a W(\frac{t}{a^2})$ is also a Brownian motion.
If W is a wiggly curve, when we zoom in billions of times, it is still very wiggly. This means there's no tangent line approximating the curvature of the local curve.
Consider
$$ Z = \frac{W(t+\Delta) - W(t)}{\Delta} $$
$Var(Z) = 1/\Delta$, which becomes larger when $\Delta\to0$. So the limitation does not exist.
