Under the terms of the Creative Commons Attribution License 3.0 (CC-BY). ; We investigate static correlation and delocalization errors in the self-consistent GW and random-phase approximation (RPA) by studying molecular dissociation of the H2 and LiH molecules. Although both approximations contain topologically identical diagrams, the nonlocality and frequency dependence of the GW self-energy crucially influence the different energy contributions to the total energy as compared to the use of a static local potential in the RPA. The latter leads to significantly larger correlation energies, which allow for a better description of static correlation at intermediate bond distances. The substantial error found in GW is further analyzed by comparing spin-restricted and spin-unrestricted calculations. At large but finite nuclear separation, their difference gives an estimate of the so-called fractional spin error normally determined only in the dissociation limit. Furthermore, a calculation of the dipole moment of the LiH molecule at dissociation reveals a large delocalization error in GW making the fractional charge error comparable to the RPA. The analyses are supplemented by explicit formulas for the GW Green's function and total energy of a simplified two-level model providing additional insights into the dissociation limit. ; AR acknowledges financial support by the European Research Council Advanced Grant DYNamo (ERC-2010-AdG-267374), European Commission project CRONOS (Grant number 280879-2 CRONOS CP-FP7), Spanish Grant (FIS2013-46159-C3-1-P) and Grupo Consolidado UPV/EHU del Gobierno Vasco (IT578-13). PR acknowledges financial support from the Academy of Finland through its Centres of Excellence Program (Project No. 251748). ; Peer Reviewed