Anomaly detection (AD) has attracted remarkable attention in hyperspectral image (HSI) processing fields, and most existing deep learning (DL)-based algorithms indicate dramatic potential for detecting anomaly samples through specific training process under current scenario. However, the limited prior information and the catastrophic forgetting problem indicate crucial challenges for existing DL structure in open scenarios cross-domain detection. In order to improve the detection performance, a novel continual learning-based capsule differential generative adversarial network (CL-CaGAN) is proposed to elevate the cross-scenario learning performance for facilitating the real application of DL-based structure in hyperspectral AD (HAD) task. First, a modified capsule structure with adversarial learning network is constructed to estimate the background distribution for surmounting the deficiency of prior information. To mitigate the catastrophic forgetting phenomenon, clustering-based sample replay strategy and a designed extra self-distillation regularization are integrated for merging the history and future knowledge in continual AD task, while the discriminative learning ability from previous detection scenario to current scenario is retained by the elaborately designed structure with continual learning (CL) strategy. In addition, the differentiable enhancement is enforced to augment the generation performance of the training data. This further stabilizes the training process with better convergence and efficiently consolidates the reconstruction ability of background samples. To verify the effectiveness of our proposed CL-CaGAN, we conduct experiments on several real HSIs, and the results indicate that the proposed CL-CaGAN demonstrates higher detection performance and continuous learning capacity for mitigating the catastrophic forgetting under cross-domain scenarios.