This paper presents a comprehensive comparison and analysis of shallow and deep learning methods within the field of machine learning. We explore the key concepts, advantages, and limitations of both approaches, focusing on their application to various data types and problem complexities. Through a qualitative analysis, we propose a theoretical framework to determine when deep learning is preferable to shallow learning, based on the symmetry properties of the corresponding function class. We further validate this framework through a case study involving hotel booking data, where we compare the prediction accuracy of five machine learning methods, including Logistic Regression, Random Forest, Recurrent Neural Network, Deep Neural Network, and Convolutional Neural Network. The results demonstrate that the proposed framework provides a valuable guidance for method selection in practical applications. Additionally, the case study highlights the positive and negative aspects of each method, offering insights into their respective performance and applicability. This research contributes to the understanding of shallow and deep learning methods and provides a practical tool for selecting the most appropriate approach based on specific problem characteristics.