Density functional theory (DFT) with a tailored Hartree-Fock hybrid functional, which can overcome the band gap problem arising in conventional DFT and gives a valence band width comparable with experiment, is applied to determine formation energies and electronic structures of intrinsic defects in cubic silicon carbide (3C-SiC). Systematic comparison of defect formation energies obtained with the tailored hybrid functional and a conventional DFT functional clearly demonstrates that conventional DFT results are not satisfactory. The understanding on intrinsic defects, which were previously investigated mainly with conventional DFT functionals, is largely revised with regard to formation energies, electronic structures and transition levels. It is found that conventional DFT functionals basically lead to (i) underestimation of the formation energy when the defect charge is more negative and (ii) overestimation when the defect charge is more positive. The underestimation is mainly attributed to the well-known band gap problem. The overestimation is attributed to shrinkage of the valence bands, although in some cases such band shrinkage may lead to underestimation depending on how the defect alters the valence band structure. Both the band gap problem and the valence band shrinkage are often observed in semiconductors, including SiC, with conventional DFT functionals, and thus need to be carefully dealt with to achieve reliable computational results.