  The distributions of Compton and Thomson radiation for a shaped laser pulse
colliding with a free electron are calculated in the framework of quantum and
classical electrodynamics, respectively. We introduce a scaling law for the
Compton and the Thomson frequency distributions which universally applies to
long and short incident pulses. Thus, we extend the validity of frequency
scaling postulated in previous studies comparing nonlinear Compton and Thomson
processes. The scaling law introduced in this paper relates the Compton no-spin
flipping process to the Thomson process over nearly the entire spectrum of
emitted radiation, including its high-energy portion. By applying the frequency
scaling, we identify that both spin and polarization effects are responsible
for differences between classical and quantum results. The same frequency
scaling applies to angular distributions and to temporal power distributions of
emitted radiation, which we illustrate numerically.
