As we know, classic derivative $f'(x)$ of a function $f(x)$ can be interpreted as the rate of change of function $f$ in each point $x.$
How about weak derivative? Since it is defined through integral and therefore not relevant in sets of zero measure, what does it mean for a function to have weak derivative? What can the weak derivative of a function explains about its function?
If $g$ is the weak derivative of $f$, then the Fundamental Theorem of Calculus holds: $$f(b)-f(a) = \int_a^b g(x)\,dx \tag{1}$$ Thus, even though $g$ may not succeed in expressing the infinitesimal rate of change of $f$ at every point, it captures the rate of change of $f$ on every non-infinitesimal scale.
For the purpose of estimating $f$, property $(1)$ is about as good as having classical derivative. For example, if $g$ happens to be square-integrable, we can write $$|f(b)-f(a)| \le \int_a^b |g(x)|\,dx \le\sqrt{b-a} \sqrt{\int_a^b |g(x)|^2\,dx}$$ and conclude that $f$ is Hölder continuous with exponent $1/2$.
For some other purposes, like locating the maximum or minimum of $f$, the weak derivative is not the right tool. For that one would use another generalized concept of derivative (Subderivative).