The usage of deep neural networks in critical systems is limited by our ability to guarantee their correct behavior. Runtime monitors are components aiming to identify unsafe predictions before they can lead to catastrophic consequences. Several recent works on runtime monitoring have focused on out-of-distribution (OOD) detection, i.e., identifying inputs that are different from the training data. In this work, we argue that OOD detection is not a well-suited framework to design efficient runtime monitors and that it is more relevant to evaluate monitors based on their ability to discard incorrect predictions. We discuss the conceptual differences with OOD and conduct extensive experiments on popular datasets to show that: 1. good OOD results can give a false impression of safety, 2. comparison under the OOD setting does not allow identifying the best monitor to detect errors.