Mild Traumatic Brain Injury (mTBI) is a common and challenging condition to diagnose accurately. Timely and precise diagnosis is essential for effective treatment and improved patient outcomes. Traditional diagnostic methods for mTBI often have limitations in terms of accuracy and sensitivity. In this study, we introduce an innovative approach to enhance mTBI diagnosis using 3D Computed Tomography (CT) images and a metric learning technique trained with triplet loss. To address these challenges, we propose a Residual Triplet Convolutional Neural Network (RTCNN) model to distinguish between mTBI cases and healthy ones by embedding 3D CT scans into a feature space. The triplet loss function maximizes the margin between similar and dissimilar image pairs, optimizing feature representations. This facilitates better context placement of individual cases, aids informed decision-making, and has the potential to improve patient outcomes. Our RTCNN model shows promising performance in mTBI diagnosis, achieving an average accuracy of 94.3%, a sensitivity of 94.1%, and a specificity of 95.2%, as confirmed through a five-fold cross-validation. Importantly, when compared to the conventional Residual Convolutional Neural Network (RCNN) model, the RTCNN exhibits a significant improvement, showcasing a remarkable 22.5% increase in specificity, a notable 16.2% boost in accuracy, and an 11.3% enhancement in sensitivity. Moreover, RTCNN requires lower memory resources, making it not only highly effective but also resource-efficient in minimizing false positives while maximizing its diagnostic accuracy in distinguishing normal CT scans from mTBI cases. The quantitative performance metrics provided and utilization of occlusion sensitivity maps to visually explain the model's decision-making process further enhance the interpretability and transparency of our approach.