In the classic Databricks architecture, the control plane hosts the Databricks web application, which is the management interface for users to create and manage clusters, jobs, notebooks, and other resources. The control plane manages the infrastructure and orchestration, while the data plane (worker nodes, driver nodes, and Databricks Filesystem) runs in customer's cloud account. Worker nodes and driver nodes are part of the data plane where actual data processing occurs, and Databricks Filesystem (DBFS) is also part of the data plane storage layer.