Speech pathology has impacts on communication abilities and quality of life. While deep learning-based models have shown potential in diagnosing these disorders, the use of sensitive data raises critical privacy concerns. Although differential privacy (DP) has been explored in the medical imaging domain, its application in pathological speech analysis remains largely unexplored despite the equally critical privacy concerns. To the best of our knowledge, this study is the first to investigate DP's impact on pathological speech data, focusing on the trade-offs between privacy, diagnostic accuracy, and fairness. Using a large, real-world dataset of 200 hours of recordings from 2,839 German-speaking participants, we observed a maximum accuracy reduction of 3.85% when training with DP with high privacy levels. To highlight real-world privacy risks, we demonstrated the vulnerability of non-private models to gradient inversion attacks, reconstructing identifiable speech samples and showcasing DP's effectiveness in mitigating these risks. To explore the potential generalizability across languages and disorders, we validated our approach on a dataset of Spanish-speaking Parkinson's disease patients, leveraging pretrained models from healthy English-speaking datasets, and demonstrated that careful pretraining on large-scale task-specific datasets can maintain favorable accuracy under DP constraints. A comprehensive fairness analysis revealed minimal gender bias at reasonable privacy levels but underscored the need for addressing age-related disparities. Our results establish that DP can balance privacy and utility in speech disorder detection, while highlighting unique challenges in privacy-fairness trade-offs for speech data. This provides a foundation for refining DP methodologies and improving fairness across diverse patient groups in real-world deployments.