We present results on transverse momentum (pT) and rapidity (y) differential production cross sections, mean transverse momentum and mean transverse momentum square of inclusive J / ψ and ψ(2 S) at forward rapidity (2.5 < y< 4) as well as ψ(2 S) -to-J / ψ cross section ratios. These quantities are measured in pp collisions at center of mass energies s=5.02 and 13 TeV with the ALICE detector. Both charmonium states are reconstructed in the dimuon decay channel, using the muon spectrometer. A comprehensive comparison to inclusive charmonium cross sections measured at s=2.76, 7 and 8 TeV is performed. A comparison to non-relativistic quantum chromodynamics and fixed-order next-to-leading logarithm calculations, which describe prompt and non-prompt charmonium production respectively, is also presented. A good description of the data is obtained over the full pT range, provided that both contributions are summed. In particular, it is found that for pT>15 GeV/c the non-prompt contribution reaches up to 50% of the total charmonium yield.