Spiking Neural Networks (SNNs) have gained huge attention as a potential energy-efficient alternative to conventional Artificial Neural Networks (ANNs) due to their inherent high-sparsity activation. Recently, SNNs with backpropagation through time (BPTT) have achieved a higher accuracy result on image recognition tasks than other SNN training algorithms. Despite the success from the algorithm perspective, prior works neglect the evaluation of the hardware energy overheads of BPTT due to the lack of a hardware evaluation platform for this SNN training algorithm. Moreover, although SNNs have long been seen as an energy-efficient counterpart of ANNs, a quantitative comparison between the training cost of SNNs and ANNs is missing. To address the aforementioned issues, in this work, we introduce SATA (Sparsity-Aware Training Accelerator), a BPTT-based training accelerator for SNNs. The proposed SATA provides a simple and re-configurable systolic-based accelerator architecture, which makes it easy to analyze the training energy for BPTT-based SNN training algorithms. By utilizing the sparsity, SATA increases its computation energy efficiency by $5.58 \times$ compared to the one without using sparsity. Based on SATA, we show quantitative analyses of the energy efficiency of SNN training and compare the training cost of SNNs and ANNs. The results show that, on Eyeriss-like systolic-based architecture, SNNs consume $1.27\times$ more total energy with sparsities when compared to ANNs. We find that such high training energy cost is from time-repetitive convolution operations and data movements during backpropagation. Moreover, to propel the future SNN training algorithm design, we provide several observations on energy efficiency for different SNN-specific training parameters and propose an energy estimation framework for SNN training. Code for our framework is made publicly available.