Radical Computational Calculations
Disclaimer: this is a preliminary answer based on the current state of the discussion with OP; it might be the subject to major changes in the near future.

So.


OP (original poster) is trying to reproduce some relative energies previously computed at ωB97X-D/6-31+G(d,p) at a different M06-2X/6-31G(d,p) level of theory. OP has exact same results for closed-shell molecular species in question, the meaning of which is probably that the resulting M06-2X/6-31G(d,p) relative energies in kcal/mol or kJ/mol are exactly equal to the previously published ωB97X-D/6-31+G(d,p).
OP also mentioned that he has some troubles in reproducing the relative energies for open-shell species, by which he probably means that for open-shell species M06-2X/6-31G(d,p) spits out different numbers in kcal/mol or kJ/mol than ωB97X-D/6-31+G(d,p).


Before we have some numbers let me say few things which hold in generality.


The difference in few kJ/mol in relative energies computed using two different functionals is pretty normal (even at exact same geometry). One should not even expect exactly same relative energies with any two different functionals in the first place.
OP presumably run geometry optimization at M06-2X/6-31G(d,p) but did not mention how different are the equilibrium geometries obtained at this level from those at ωB97X-D/6-31+G(d,p) level published previously. In fact, this would be difficult to do, because I could not find the Cartesian coordinates of the species in question in the Supporting Information for the work published previously. But at least some quick visual comparison has to be done to make sure that equilibrium geometries are at least not totally different.
OP did not mention why M06-2X functional was chosen. Even if the ultimate goal is not just to reproduce the previously published work, it is mentioned there that:


  ωB97X-D, a range-separated double-hybrid functional, has been shown to describe reaction energies accurately.45


But, first of all, ωB97X-D is not a double-hybrid functional. And secondly, the paper quoted there is fairly well-known benchmark of density functional methods done by Lars Goerigkab and Stefan Grimme, and in fact, Goerigkab and Grimme found M062X-D3 and not ωB97X-D to be "statistically the best of all hybrids". Note though that M062X-D3 differs from M062X: the former includes the empirical dispersion correction while the later does not.
With respect to the empirical dispersion correction, today it is usually recommended to include it in DFT calculations, since it is available at almost no (computational) cost but can substantially improve the results. Note also that dispersion correction is already included in case of ωB97X-D functional but has to be "manually" added for the case of M06-2X. In fact, the absence of it might explain some differences in equilibrium geometries and relative energies.