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Differentiable simulators provide an avenue for closing the sim-to-real gap by enabling the use of efficient, gradient-based optimization algorithms to find the simulation parameters that best fit the observed sensor readings. Nonetheless, these analytical models can only predict the dynamical behavior of systems for which they have been designed. In this work, we study the augmentation of a novel differentiable rigid-body physics engine via neural networks that is able to learn nonlinear relationships between dynamic quantities and can thus learn effects not accounted for in traditional simulators.Such augmentations require less data to train and generalize better compared to entirely data-driven models. Through extensive experiments, we demonstrate the ability of our hybrid simulator to learn complex dynamics involving frictional contacts from real data, as well as match known models of viscous friction, and present an approach for automatically discovering useful augmentations. We show that, besides benefiting dynamics modeling, inserting neural networks can accelerate model-based control architectures. We observe a ten-fold speed-up when replacing the QP solver inside a model-predictive gait controller for quadruped robots with a neural network, allowing us to significantly improve control delays as we demonstrate in real-hardware experiments. We publish code, additional results and videos from our experiments on our project webpage at https://sites.google.com/usc.edu/neuralsim.