Pneumonia, particularly when induced by diseases like COVID-19, remains a critical global health challenge requiring rapid and accurate diagnosis. This study presents a comprehensive comparison of traditional machine learning and state-of-the-art deep learning approaches for automated pneumonia detection using chest X-rays (CXRs). We evaluate multiple methodologies, ranging from conventional machine learning techniques (PCA-based clustering, Logistic Regression, and Support Vector Classification) to advanced deep learning architectures including Convolutional Neural Networks (Modified LeNet, DenseNet-121) and various Vision Transformer (ViT) implementations (Deep-ViT, Compact Convolutional Transformer, and Cross-ViT). Using a dataset of 5,856 pediatric CXR images, we demonstrate that Vision Transformers, particularly the Cross-ViT architecture, achieve superior performance with 88.25% accuracy and 99.42% recall, surpassing traditional CNN approaches. Our analysis reveals that architectural choices impact performance more significantly than model size, with Cross-ViT's 75M parameters outperforming larger models. The study also addresses practical considerations including computational efficiency, training requirements, and the critical balance between precision and recall in medical diagnostics. Our findings suggest that Vision Transformers offer a promising direction for automated pneumonia detection, potentially enabling more rapid and accurate diagnosis during health crises.