Machine Learning (ML) models integrated with in-situ sensing offer transformative solutions for defect detection in Additive Manufacturing (AM), but this integration brings critical challenges in safeguarding sensitive data, such as part designs and material compositions. Differential Privacy (DP), which introduces mathematically controlled noise, provides a balance between data utility and privacy. However, black-box Artificial Intelligence (AI) models often obscure how this noise impacts model accuracy, complicating the optimization of privacy-accuracy trade-offs. This study introduces the Differential Privacy-Hyperdimensional Computing (DP-HD) framework, a novel approach combining Explainable AI (XAI) and vector symbolic paradigms to quantify and predict noise effects on accuracy using a Signal-to-Noise Ratio (SNR) metric. DP-HD enables precise tuning of DP noise levels, ensuring an optimal balance between privacy and performance. The framework has been validated using real-world AM data, demonstrating its applicability to industrial environments. Experimental results demonstrate DP-HD's capability to achieve state-of-the-art accuracy (94.43%) with robust privacy protections in anomaly detection for AM, even under significant noise conditions. Beyond AM, DP-HD holds substantial promise for broader applications in privacy-sensitive domains such as healthcare, financial services, and government data management, where securing sensitive data while maintaining high ML performance is paramount.