Open-World Continual Learning (OWCL) aims to simulate real-world scenarios where tasks evolve over time, category distributions shift dynamically, and the model encounters previously unseen (unknown) samples. The learning system is expected not only to retain knowledge of previously learned tasks while acquiring new ones, but also to accurately identify unknown categories, thereby achieving continuous and robust knowledge accumulation and generalization. However, most existing continual learning approaches are built upon the closed-world assumption and struggle to cope with the uncertainty brought by open categories and inter-task interference. In particular, they exhibit clear limitations in balancing knowledge stability and plasticity. To address these challenges, this paper formalizes the OWCL problem and proposes a Task-aware Prompt-driven Mixture-of-Experts model (TP-MoE) that dynamically models task semantics and efficiently routes expert modules to support knowledge transfer and continual knowledge update. Specifically, TP-MoE incorporates a plug-and-play prompt aggregation mechanism and improves the expert gating strategy for better routing based on task semantics. It also introduces an adaptive open-set thresholding strategy, which dynamically adjusts decision boundaries based on knowledge evolution, enhancing both open-category detection and known-category classification. Experimental results on two public benchmarks demonstrate that TP-MoE consistently outperforms existing methods across various metrics, exhibiting superior robustness and generalization. This work provides a scalable and transferable framework for knowledge modeling and task scheduling in open-world continual learning.