Here I will expand my comment, which to me is a very natural distinction between root-mean-square vs. mean-absolute deviations.
Say you have some data $x_1,\ldots,x_n$, which you want to approximate by a constant $c$, i.e. $$x_i\approx c$$ for all $i$.
How do you choose the constant? A common approach is to minimize some error $E[c]$.
One choice for $E$ is the sum square error $$E_\text{SSE}=\sum_i\big(x_i-c\big)^2$$ the solution will then be $c_\min=\frac{1}{n}\sum x_i$. In other words, we have $$\big[c_\min,E_\min\big]_\text{SSE}=\big[\text{mean}(\mathbf{x}),n\,\text{var}(\mathbf{x})\big]$$ so if you are using the mean as your measure of central tendency, the RMS error is really the "natural" measure of dispersion.
On the other hand, if we choose $E$ to be the sum absolute error $$E_\text{SAE}=\sum_i\big|x_i-c\big|$$ the solution will then be $(c_\min)_\text{SAE}=\text{median}(\mathbf{x})$. So if you want to use mean absolute deviation to measure dispersion, really the "natural" measure of central tendency would be the median.