Speech analysis for clinical applications has emerged as a burgeoning field, providing valuable insights into an individual's physical and physiological state. Researchers have explored speech features for clinical applications, such as diagnosing, predicting, and monitoring various pathologies. Before presenting the new deep learning frameworks, this thesis introduces a study on conventional acoustic feature changes in subjects with post-traumatic headache (PTH) attributed to mild traumatic brain injury (mTBI). This work demonstrates the effectiveness of using speech signals to assess the pathological status of individuals. At the same time, it highlights some of the limitations of conventional acoustic and linguistic features, such as low repeatability and generalizability. Two critical characteristics of speech features are (1) good robustness, as speech features need to generalize across different corpora, and (2) high repeatability, as speech features need to be invariant to all confounding factors except the pathological state of targets. This thesis presents two research thrusts in the context of speech signals in clinical applications that focus on improving the robustness and repeatability of speech features, respectively. The first thrust introduces a deep learning framework to generate acoustic feature embeddings sensitive to vocal quality and robust across different corpora. A contrastive loss combined with a classification loss is used to train the model jointly, and data-warping techniques are employed to improve the robustness of embeddings. Empirical results demonstrate that the proposed method achieves high in-corpus and cross-corpus classification accuracy and generates good embeddings sensitive to voice quality and robust across different corpora. The second thrust introduces using the intra-class correlation coefficient (ICC) to evaluate the repeatability of embeddings. A novel regularizer, the ICC regularizer, is proposed to regularize deep neural networks to produce embeddings with higher repeatability. This ICC regularizer is implemented and applied to three speech applications: a clinical application, speaker verification, and voice style conversion. The experimental results reveal that the ICC regularizer improves the repeatability of learned embeddings compared to the contrastive loss, leading to enhanced performance in downstream tasks.