The Cox proportional hazards model is often used to analyze data from Randomized Controlled Trials (RCT) with time-to-event outcomes. Random survival forest (RSF) is a machine-learning algorithm known for its high predictive performance. We conduct a comprehensive neutral comparison study to compare the performance of Cox regression and RSF in various simulation scenarios based on two reference datasets from RCTs. The motivation is to identify settings in which one method is preferable over the other when comparing different aspects of performance using measures according to the TRIPOD (Transparent Reporting of a multivariable prediction model for Individual Prognosis or Diagnosis) recommendations. Our results show that conclusions solely based on the C index, a performance measure that has been predominantly used in previous studies comparing predictive accuracy of the Cox-PH and RSF model based on real-world observational time-to-event data and that has been criticized by methodologists, may not be generalizable to other aspects of predictive performance. We found that measures of overall performance may generally give more reasonable results, and that the standard log-rank splitting rule used for the RSF may be outperformed by alternative splitting rules, in particular in nonproportional hazards settings. In our simulations, performance of the RSF suffers less in data with treatment-covariate interactions compared to data where these are absent. Performance of the Cox-PH model is affected by the violation of the proportional hazards assumption.