difference between Sobolev space and bounded variation space

I learned Sobolev spaces & bounded variation spaces. I read this sentence:

Sobolev spaces include functions such that $\int |f^{(1)}(x)|^2 dx < \infty$, but do not include functions that are piecewise smooth. The space of bounded variation $\{f: \int |f^{(1)}(x)|dx < \infty\}$ include such functions.

So the only difference in two equations is square, I am not sure how this makes a big difference. Can anyone elaborate on this difference with example of step function $f(x)=0$ if $x<0$, $f(x)=1$ if $x\ge0$ for $x \in [-1,1]$?