As near as I can tell: Compton scattering treats the photon and electron as two colliding balls, assigning momentum to the photon based on its wavelength. However, because the electron changed momentum during the collision, it must have experienced acceleration. Maxwell's eqns tells you that an accelerating charge radiates, but this radiation is not accounted for in the elementary Compton calculation. Compton got away with the elementary calculation because the effect is apparent at high energies only.
[This is the first I've heard of the effect, and I'm thinking it out in real time, so apply grain of salt.]
Radiation reaction is related to Compton scattering. In Compton scattering, the interaction is between one electron and one photon producing one photon at a new energy, with the electron changing energy. Non-linear Compton is one electron interacting with many photons to produce an electron and a high energy photon. Radiation reaction is essentially one electron scattering off many photons many times, producing many high energy photons. The paper itself is about the first observation of radiation reaction, there is work showing electrons losing energy in non-linear Compton scattering from the 1990s.
It's not. It's an inverse Compton scattering experiment. Abstract: "The generated gamma-rays have the highest energies yet reported from an all-optical inverse Compton scattering scheme..."
I do think there is a difference, but they are aspects of the same thing. One significant difference is that in the classical limit (ie taking the limit hbar -> 0) non-linear Compton scattering doesn’t produce energy loss for the electrons. The classical limit of radiation reaction (ie n photons scattering of an electrons m times) does produce energy loss (because the loss per scatter scales like hbar but the number of scatters scales like 1/hbar). The sentence you quote is about the energy of the photons produced (which determined by Compton scattering) but the energy loss of the electrons is what we call radiation reaction.
