With the evolution of information and data tech, telemedicine enables patients to access healthcare at home. Yet, complex health issues require telemedicine to complement in-person care, necessitating seamless patient data exchange between telemedicine platforms and traditional healthcare facilities. This collaboration occurs intra-hospital or inter-hospital, defining two telemedicine modalities. Using game theory, our analysis investigates how these models affect hospital competition and cooperation dynamics. Surprisingly, implementing telemedicine doesn't guarantee increased profitability; rather, it hinges on model specifics. Intra-hospital telemedicine can ease competition under particular circumstances, fostering mutual gains, albeit with stringent requirements. Conversely, inter-hospital telemedicine may present a win-win scenario, with heightened profitability for both under suitable conditions. Our research highlights the subtleties of telemedicine's economic impact on hospitals, underscoring that benefits depend critically on context and strategic planning.