Continual Learning (CL) models, while adept at sequential knowledge acquisition, face significant and often overlooked privacy challenges due to accumulating diverse information. Traditional privacy methods, like a uniform Differential Privacy (DP) budget, indiscriminately protect all data, leading to substantial model utility degradation and hindering CL deployment in privacy-sensitive areas. To overcome this, we propose a privacy-enhanced continual learning (PeCL) framework that forgets what's sensitive and remembers what matters. Our approach first introduces a token-level dynamic Differential Privacy strategy that adaptively allocates privacy budgets based on the semantic sensitivity of individual tokens. This ensures robust protection for private entities while minimizing noise injection for non-sensitive, general knowledge. Second, we integrate a privacy-guided memory sculpting module. This module leverages the sensitivity analysis from our dynamic DP mechanism to intelligently forget sensitive information from the model's memory and parameters, while explicitly preserving the task-invariant historical knowledge crucial for mitigating catastrophic forgetting. Extensive experiments show that PeCL achieves a superior balance between privacy preserving and model utility, outperforming baseline models by maintaining high accuracy on previous tasks while ensuring robust privacy.