Least-squares fitting minimises the sum of SQUARED residuals rather than the sum of absolute ones. What does squaring change, and what does it cost?

A. It gives a closed-form solution, at the cost of weighting outliers heavily.

Squaring buys a smooth convex objective solvable in closed form through the normal equations, where absolute error has a corner at zero and needs iteration. The cost is sensitivity: a residual of 10 contributes 100, so the fit works disproportionately hard on large errors. Absolute values also prevent cancellation, so that is not what distinguishes them, the trade is exactness against robustness.