The role of many-electron effects in the formation of electronic quasiparticle spectra in organic molecular semiconductors (OMS) is analyzed. Many-body perturbation theory, ab initio calculations of metal phthalocyanines and PTCDA molecules, and experimental photoemission spectra are applied to this analysis. It is shown that density functional theory (DFT) poorly reproduces the electronic spectra of OMS. The use of a hybrid functional method (HFM) provides precise reproduction of both valence and conducting bands, while the HOMO-LUMO gap remains underestimated. The correct gap width is obtained in both DFT and HFM, when it is calculated through ionization and affinity energies. It is shown that such an approach gives a formula for gap correction due to electron correlations, which is close to an expression derived from the GW approximation.