The PMBOK was launched by PMI in 1983. PMI offers a certificate examination by which one can become a certified Project Management Professional (PMP). Project managers with the PMP credential indicate that they can direct and manage a project by following the processes and knowledge areas embodied in the PMBOK.
The PMBOK is a collection of processes and knowledge areas that are generally accepted as the best practices within the discipline of project management. The PMBOK structures project management processes into five basic groups known as process groups: initiating, planning, executing, monitoring and controlling, and closing. These Project Management Process Groups also comprise 42 management processes that describe activities throughout a project’s life cycle. These management processes include the tools and techniques used for applying knowledge and skills described in the nine knowledge areas, namely project integration management, project scope management, project time management, project cost management, project quality management, project human resource management, project communication management, project risk management and project procurement management.
The PMBOK is the standard for managing projects across various industries, including software projects, and is mostly used in US-based organisations.
So what can I summarize is that PMI is an organization meanwhile PMBOK is a framework/standard but not a project management methodology.