Proton computed tomography (pCT) has been proposed as an alternative to x-ray computed tomography (CT) for acquiring relative to water stopping power (RSP) maps used for proton treatment planning dose calculations. In parallel, it has been shown that dual energy x-ray CT (DECT) improves RSP accuracy when compared to conventional single energy x-ray CT. This study aimed at directly comparing the RSP accuracy of both modalities using phantoms scanned at an advanced prototype pCT scanner and a state-of-the-art DECT scanner. Two phantoms containing 13 tissue-mimicking inserts of known RSP were scanned at the pCT phase II prototype and a latest generation dual-source DECT scanner (Siemens SOMATOM Definition FORCE). RSP accuracy was compared by mean absolute percent error (MAPE) over all inserts. A highly realistic Monte Carlo (MC) simulation was used to gain insight on pCT image artifacts which degraded MAPE. MAPE was 0.55% for pCT and 0.67% for DECT. The realistic MC simulation agreed well with pCT measurements (MAPE = 0.69%). Both simulation and experimental results showed ring artifacts in pCT images which degraded the MAPE compared to an ideal pCT simulation (MAPE = 0.17%). Using the realistic simulation, we could identify sources of artifacts, which are attributed to the interfaces in the five-stage plastic scintillator energy detector and calibration curve interpolation regions. Secondary artifacts stemming from the proton tracker geometry were also identified. The pCT prototype scanner outperformed a state-of-the-art DECT scanner in terms of RSP accuracy (MAPE) for plastic tissue mimicking inserts. Since artifacts tended to concentrate in the inserts, their mitigation may lead to further improvements in the reported pCT accuracy.