Personnel

Definition: People employed in an organization or engaged in an organized undertaking such as military service. (Oxford Dictionary)

Example: There are little villages set up in the training area, and military personnel are tasked to play civilians.

Staff

Definition: 1. All the people employed by a particular organization. 2. A group of officers assisting an officer in command of an army formation or administration headquarters.(Oxford Dictionary)

Example: Up to 12 members of staff have left the college because they are unhappy with the new setup.

As we can see from their definitions, they have the same meaning when people employed in an organization are referred to. However, personnel also include all people in military service, while staff mainly refers to employees in an organization (school administrators, a group of senior army officers who help a commanding officer, etc.).