Azure DevOps Services is a cloud-hosted platform that provides scalable and globally available services with a high SLA. Azure DevOps Server is installed on-premises and runs on SQL Server, allowing customers to keep their data within their network. Key differences include simplified management and automatic updates with Services, while Server provides access to SQL Server reporting and on-premises data storage. Customers should consider their data security, management, and reporting needs to determine which platform best fits their requirements.