Sometimes $\mathbb{E}[X^2]$ just seems handier than merely using $\mathbb{E}[|X|]$? Why?
I'll give an example. An exercise in our class was this: Let $(X_k)$ be a sequence of independent random variables with $$ \mathbb{P}[X_k = k] = \mathbb{P}[X_k = -k] = \frac{1}{2k \log(k+1)}, \ \mathbb{P}[X_k = 0] = 1 - \frac{1}{k \log(k+1)}. $$ Show that $\frac{1}{n} \sum_{k=1}^n X_k \xrightarrow{\mathbb{P}} 0$.
One could aim to apply Chebyshev/Markov to get $$ \mathbb{P} \left[ \left|\frac{1}{n} \sum_{k=1}^n X_k \right| > \varepsilon \right] \leq \frac{\mathbb{E} \left[ \left|\frac{1}{n}\sum_{k=1}^n X_k \right|\right]}{\varepsilon}.$$ But this does not help too much (at least not immediately).
If we however instead bound it in $L^2$, then we can apply linearity of $\operatorname{Var}$, etc. and relatively quickly resolve this exercise.
I've seen this happen in several other occasions. So why is the second moment sometimes helpful - or better: when (and why!) is it more helpful? The variance describes different things than the expected value but e.g. in this case it's "just" the difference of $\mathbb{E}[|X|]$ and $\mathbb{E}[X^2]$.
Is it simply because $L^2$ is a Hilbert Space and hence naturally, everything is much better?