This work contributes an event-driven visual-tactile perception system, comprising a novel biologically-inspired tactile sensor and multi-modal spike-based learning. Our neuromorphic fingertip tactile sensor, NeuTouch, scales well with the number of taxels thanks to its event-based nature. Likewise, our Visual-Tactile Spiking Neural Network (VT-SNN) enables fast perception when coupled with event sensors. We evaluate our visual-tactile system (using the NeuTouch and Prophesee event camera) on two robot tasks: container classification and rotational slip detection. On both tasks, we observe good accuracies relative to standard deep learning methods. We have made our visual-tactile datasets freely-available to encourage research on multi-modal event-driven robot perception, which we believe is a promising approach towards intelligent power-efficient robot systems.