Image classification and segmentation are common applications of deep learning to radiology. While many tasks can be framed using either classification or segmentation, classification has historically been cheaper to label and more widely used. However, recent work has drastically reduced the cost of training segmentation networks. In light of this recent work, we reexamine the choice of training classification vs. segmentation models. First, we use an information theoretic approach to analyze why segmentation vs. classification models may achieve different performance on the same dataset and overarching task. We then implement multiple methods for using segmentation models to classify medical images, which we call segmentation-for-classification, and compare these methods against traditional classification on three retrospective datasets. We use our analysis and experiments to summarize the benefits of switching from segmentation to classification, including: improved sample efficiency, enabling improved performance with fewer labeled images (up to an order of magnitude lower), on low-prevalence classes, and on certain rare subgroups (up to 161.1\% improved recall); improved robustness to spurious correlations (up to 44.8\% improved robust AUROC); and improved model interpretability, evaluation, and error analysis.