Derby is an embedded database and is great for self-contained installations where a separate database server is not available. CM1 can be installed using the embedded Derby database, leading to a simplified setup. The impact of this is that Derby runs in the CM1 process and on the same machine as CM1. This means that memory available for CM1 and the CPU on the CMS server will be used for the database work, CM1 UI, and publishing work.
For organizations with larger amounts of content and enterprise level operations, a separate database can improve performance and user experience. If you have access to a separate database server, are publishing frequently, and maintain a larger number of users actively in the system, MySQL would is a more scalable option.