With standard deviation as defined, you get cool results like Chebyshev's Theorem: for any distribution and k>1, at most 1/k^2 of the data fall outside of k standard deviations from the mean.  So, for example, for any distribution at most a quarter of the data lies farther then two standard deviations from the mean, and at most 12% lie further than three standard deviations.

This and other theoretical advantages come from the long answer that Mark4483 gave.  These things are important for developing inference models.