In real-world scenarios, continual learning is essential for adapting models to open environments, but it faces the catastrophic forgetting problem in the context of Open-World Recognition. This task, which combines continual learning with open-set methods like evidential deep learning, also requires out-of-distribution detection capabilities, further increasing the challenge of addressing catastrophic forgetting. To address this issue, we propose an enhanced continual learning approach grounded in evidential deep learning. Specifically, we introduce Logits-Angle Knowledge Distillation, a novel strategy that introduces angle as a metric that is employed to gauge the consistency of the local distribution patterns of samples within the feature spaces corresponding to the preceding model and the current model. This measure contributes to maintaining the generalization ability of the current model on the previously learned classes. In addition, we introduce Weighted Evidential Deep Learning to address the sample imbalance between old and new classes. This method adaptively adjusts a balancing factor based on sample counts, helping to mitigate the catastrophic forgetting problem by countering the effects of imbalance. Experimental results show that our proposed method achieves significant performance improvements on CIFAR-100 and Tiny-ImageNet: the Average Class Accuracy (ACA) is improved by at least 3.61% and 9.54%, respectively, and the Average Incremental Accuracy (AIA) is enhanced by 2.28% and 8.72%, respectively.