The self-interaction correction of Perdew and Zunger with the optimized effective potential using the Krieger-Li-Iafrate approximation is analyzed for atomic and molecular systems in the exchange-only context. Including the self-interaction correction (SIC) orbital by orbital shows that the appropriate asymptotic behavior of the exchange potential can be achieved if just the contribution of the highest occupied molecular orbital (HOMO) is considered. However, if a good description of the exchange potential in the valence region is required, and consequently a good description of the HOMO energy, then all electrons of the valence shell must be taken into account. In contrast, the lowest unoccupied molecular orbital (LUMO) is described adequately if just the HOMO SIC contribution is employed. In addition, if the lowest occupied orbital is also considered in the SIC approximation, there is an improvement in the description of the exchange potential in inner regions of an atom. When all electrons in an atom or in a molecule are considered in the SIC approximation, there exists a linear relationship between their occupied orbital energies and those obtained with the local density approximation (LDA). This indicates that the SIC and LDA occupied orbital energies are related by a shift. Furthermore, for a set of atoms or molecules there is a linear relationship between HOMO energies obtained with LDA and those obtained with the SIC approximation. Using both observations, the shift between the occupied orbital energies LDA and SIC is reported. We found that this shift (obtained for the occupied orbitals) cannot be applied to the virtual orbitals, in particular to the LUMO. However, we do find an additional linear relationship between LUMO energies obtained with LDA and those obtained with the SIC approximation. The difference between the LUMO and HOMO energy (GAP) obtained with the LDA and SIC approximations is compared with that obtained with an exact local-multiplicative exchange potential and all are compared with experimental vertical excitation energies. Whereas the LDA GAP underestimates the excitation energies, the GAP obtained with the SIC approximation and with an exact local-multiplicative exchange potential overestimates this quantity. From an analysis of the exchange energy for simple molecules, and with a similar approach to the modified X-alpha method, we found a linear relationship between the SIC and Hartree-Fock (HF) methods. We show numerically that the nondiagonal terms of the exact orbital representation of the exchange energy can be approximated by the SIC approach.