Why Isn't This Outlier Score/Reconstruction Error Not Squared?
there is a good explaination squarred vs flat.

But from the simple view squarred error aims at variance while flat at median.
Different outliers comes from both versions, this is task dependent - the choice has been made by authors.

In fact yes, it is called roboust (somehow missused term), but it is not "better" for time series, just gives better estimator for outliers in presented method, which from given equation uses neither mean error nor mean square error - it uses custom made estimator with variance calculated on it's own.