The Gini coefficient is invariant to scale and is bounded, the standard deviation invariant to a shift, and unbounded, so they are difficult to compare directly. Now you can define a scale-invariant version of the standard deviation, by dividing by the mean (coefficient of variation).
However, the Gini index is still based on values, the second on squared values, so you can expect the second one will to more influenced by outliers (excessively low or high values). This can be found in Income inequality measures, F De Maio, 2007:
This measure of income inequality is calculated by the dividing the standard deviation of the income distribution by its mean. More equal income distributions will have smaller standard deviations; as such, the CV will be smaller in more equal societies. Despite being one of the simplest measures of inequality, use of the CV has been fairly limited in the public health literature and it has not featured in research on the income inequality hypothesis. This may be attributed to important limitations of the CV measure: (1) it does not have an upper bound, unlike the Gini coefficient,18 making interpretation and comparison somewhat more difficult; and (2) the two components of the CV (the mean and the standard deviation) may be exceedingly influenced by anomalously low or high income values. In other words, the CV would not be an appropriate choice of income inequality measure if a study's income data did not approach a normal distribution.
So the coefficient of variation is less robust, and still unbounded. To take a further step, you can remove the mean, and divide by the absolute deviation instead ($\ell_1(x-m)=\sum |x_n -m|$). Up to a factor, you end up with a $\ell_1/\ell_2$ norm ratio, which can be bounded, since, for an $N$-point vector, $\ell_2(x)\le \ell_1(x)\le \sqrt{N}\ell_2(x) $.
Now you have, with the Gini index and the $\ell_1/\ell_2$ norm ratio, two interesting measures of distribution sparsity, scale-invariant and bounded.
They are compared in Comparing Measures of Sparsity, 2009. Tested against different natural sparsity properties (Robin Hood, Scaling, Rising Tide, Cloning, Bill Gates, and Babies), the Gini index stands out as the best. But its shape makes it difficult to use as a loss function, and regularized versions of the $\ell_1/\ell_2$ can be used in this context.
So unless you want to characterize a nearly Gaussian distribution, if you want to measure a sparsity, use the Gini index, if you want to promote sparsity among different models, you can try such a norm ratio.
Additional lecture: The GMD (Gini’s Mean difference): A Superior Measure of Variability for Non-Normal Distributions, Shlomo Yitzhaki, 2002, whose abstract might appear of interest:
Of all measures of variability, the variance is by far the most popular. This paper argues that Gini’s Mean Difference (GMD), an alternative index of variability, shares many properties with the variance, but can be more informative about the properties of distributions that depart from normality