We present here the second part of a project that aims at solving the controversy on the issue of the bar effect on the radial distribution of metals in the gas-phase of spiral galaxies. In Paper I we presented a compilation of more than 2800 HII regions belonging to 51 nearby galaxies for which we derived chemical abundances and radial abundance profiles from a homogeneous methodology. In this paper we analyse the derived gas-phase radial abundance profiles of 12+log(O/H) and log(N/O), for barred and unbarred galaxies separately, and find that the differences in slope between barred and unbarred galaxies depend on galaxy luminosity. This is due to a different dependence of the abundance gradients (in dex/kpc) on luminosity for the two types of galaxies: In the galaxy sample that we consider the gradients appear to be considerably shallower for strongly barred galaxies in the whole luminosity range, while profile slopes for unbarred galaxies become steeper with decreasing luminosity. Therefore, we only detect differences in slope for the lower luminosity (lower mass) galaxies (M_B >~ -19.5 or M_* <~ 10^{10.4} M_sun). We discuss the results in terms of the disc evolution and radial mixing induced by bars and spiral arms. Our results reconcile previous discrepant findings that were biased by the luminosity (mass) distribution of the sample galaxies and possibly by the abundance diagnostics employed.
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