Proof that (x-y)(x+y) is more accurate than x²-y²
Suppose $y = x + \Delta$. Then $(x^2 - y^2) = x^2 - (x^2 + 2\Delta x + \Delta^2) = - (2\Delta x + \Delta^2)$ with leading term on the order of $2 \Delta x$. Multiply by $1 + \delta_1$ and that's still the leading term. 

Compare to $(\delta_1 - \delta_2) y^2$ with leading term $(\delta_1 - \delta_2) x^2$.