Heuristic
For small $\Delta t>0$,
$$ \frac{B_{t+\Delta t}-B_t}{\Delta t} \sim N\left(0,\frac{1}{\Delta t}\right). $$Consequence
As $\Delta t\to 0$, the variance of the difference quotient blows up:
$$ \frac{1}{\Delta t}\to\infty. $$So Brownian motion paths are continuous but too rough to have an ordinary derivative.
Significance
This roughness is one reason ordinary calculus must be replaced by Itô calculus.