The main advantage of the mean-square error is the analytical tractability that comes with it. Many problems have simple solutions, either as closed-form formulas or semi-closed-form algorithms... The mathematical ease is related to the fact that problems phrased in terms of minimizing the mean-square error are equivalent to calculating projections in linear functional spaces.

On the other hand, mean absolute error ensures a much more robust estimation, insensitive to outliers and errors in the data. Applying mean absolute error in the context of variable selection allows you to substantially reduce the number of predictors you have to worry about.