Federated learning enables collaborative model training across medical institutions without sharing raw data, but its performance is often limited by domain heterogeneity across clients. Existing approaches to address this challenge fall into two main paradigms: model-side personalization, which adapts model parameters to each client, and data-side harmonization, which reduces inter-client variation at the input level. Despite their widespread use, these strategies have not been systematically compared. In this work, we conduct a comprehensive study across six medical imaging settings-colon polyp, skin lesion, and breast tumor segmentation, and tuberculosis CXR, brain tumor, and breast tumor classification-covering diverse types of domain shift. We evaluate a broad set of state-of-the-art harmonization and personalization methods under a unified framework. Our results reveal a conditional trade-off driven by the nature of heterogeneity: harmonization is more effective when variation is primarily appearance-based (e.g., CXR classification), while personalization performs better when differences are structural (e.g., colon polyp segmentation). When inter-client variation is limited, both strategies perform similarly. These findings demonstrate that the effectiveness of adaptation in federated medical imaging depends on the type and magnitude of domain shift rather than the strategy alone. We provide practical guidelines for selecting between harmonization and personalization and highlight directions for future hybrid approaches that combine both paradigms. Code is available at https://github.com/ChamaniS/WhenToAdapt.