We consider the Thomson and Compton scattering of high-energy electrons in an intense laser pulse. Our simulations show that energy losses due to radiation reaction cause the emitted radiation to be spread over a broader angular range than the case without these losses included. We explain this in terms of the effect of these energy losses on the particle dynamics. Finally, at ultra-high intensities, i.e. fields with a dimensionless parameter a0~200, the energy of the emission spectrum is significantly reduced by radiation reaction and also the classical and QED results begin to differ. This is found to be due to the classical theory overestimating the energy loss of the electrons. Such findings are relevant to radiation source development involving the next generation of high-intensity laser facilities.