Resistive switching memories (RRAMs) have attracted wide interest as adaptive synaptic elements in artificial bio-inspired spiking neural networks (SNNs). These devices suffer from high cycle-to-cycle and cell-to-cell conductance variability, which is usually considered as a big challenge. However, biological synapses are noisy devices and the brain seems in some situations to benefit from the noise. It has been predicted that RRAM-based SNNs are intrinsically robust to synaptic variability. Here, we investigate this robustness based on extensive characterization data: we analyze the role of noise during unsupervised learning by spike-timing dependent plasticity (STDP) for detection in dynamic input data and classification of static input data. Extensive characterizations of multi-kilobits HfO2-based oxide-based RAM (OxRAM) arrays under different programming conditions are presented. We identify the trade-offs between programming conditions, power consumption, conductance variability and endurance features. Finally, the experimental results are used to perform system-level simulations fully calibrated on the experimental data. The results demonstrate that, similarly to biology, SNNs are not only robust to noise but a certain amount of noise can even improve the network performance. OxRAM conductance variability increases the range of synaptic values explored during the learning process. Moreover, the reduction of constraints on the OxRAM conductance variability allows the system to operate at low power programming conditions.