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The spiking neural network (SNN) is a possible pathway for low-power and energy-efficient processing and computing exploiting spiking-driven and sparsity features of biological systems. This article proposes a sparsity-driven SNN learning algorithm, namely backpropagation with sparsity regularization (BPSR), aiming to achieve improved spiking and synaptic sparsity. Backpropagation incorporating spiking regularization is utilized to minimize the spiking firing rate with guaranteed accuracy. Backpropagation realizes the temporal information capture and extends to the spiking recurrent layer to support brain-like structure learning. The rewiring mechanism with synaptic regularization is suggested to further mitigate the redundancy of the network structure. Rewiring based on weight and gradient regulates the pruning and growth of synapses. Experimental results demonstrate that the network learned by BPSR has synaptic sparsity and is highly similar to the biological system. It not only balances the accuracy and firing rate, but also facilitates SNN learning by suppressing the information redundancy. We evaluate the proposed BPSR on the visual dataset MNIST, N-MNIST, and CIFAR10, and further test it on the sensor dataset MIT-BIH and gas sensor. Results bespeak that our algorithm achieves comparable or superior accuracy compared to related works, with sparse spikes and synapses.