Continual Novel Class Discovery (CNCD) aims to continually discover novel classes without labels while maintaining the recognition capability for previously learned classes. The main challenges faced by CNCD include the feature-discrepancy problem, the inter-session confusion problem, etc. In this paper, we propose a novel Feature Enhancement and Adaptation method for the CNCD to tackle the above challenges, which consists of a guide-to-novel framework, a centroid-to-samples similarity constraint (CSS), and a boundary-aware prototype constraint (BAP). More specifically, the guide-to-novel framework is established to continually discover novel classes under the guidance of prior distribution. Afterward, the CSS is designed to constrain the relationship between centroid-to-samples similarities of different classes, thereby enhancing the distinctiveness of features among novel classes. Finally, the BAP is proposed to keep novel class features aware of the positions of other class prototypes during incremental sessions, and better adapt novel class features to the shared feature space. Experimental results on three benchmark datasets demonstrate the superiority of our method, especially in more challenging protocols with more incremental sessions.