Climate projections for the 21st century for CMIP6 are warmer than those for CMIP5 despite nominally identical instantaneous radiative forcing. Many climate modeling groups attribute the stronger warming in the CMIP6 projections to the higher climate sensitivity of the new generation of climate models, but here we demonstrate that also changes in the forcing datasets can play an important role, in particular the prescribed concentrations of greenhouse gases (GHG) that are used to force the models. In the EC-Earth3-Veg model the effective radiative forcing (ERF) is reduced by 1.4 W m−2 when the GHG concentrations from SSP5-8.5 (used in CMIP6) are replaced by the GHG concentrations from RCP8.5 (used in CMIP5), and similar yet smaller reductions are seen for the SSP2-4.5/RCP4.5 and SSP1-2.6/RCP2.6 scenario pairs. From the reduced ERF we can estimate the temperature at the end of the century in a full climate simulation with the CMIP6 version of the EC-Earth model but using CMIP5 GHG concentrations instead. For the new SSP5-8.5 and SSP2-4.5 scenarios we find that 50% or more of the temperature increase from CMIP5 to CMIP6 at the end of the century is due to changes in the prescribed GHG concentrations. The implication is that CMIP5 and CMIP6 projections for the 21st century are difficult to compare with each other not only as models differ but also as the forcing conditions are not equal. Therefore, the communication of CMIP6 results to the impact, mitigation and adaptation communities has to be carefully formulated, taking into account the role of the updated GHG concentrations when interpreting the warmer climate projections for the 21st century.