What is the difference between a school and a department?
The term school and department are going to mean different things at different institutions. Generally speaking, the term school refers to a larger unit within a university than department and many schools will contain multiple departments. For example, Harvard University contains a School of Public Health (and other schools) which contains a Department of Epidemiology (and other departments).
That said, names are often not even used uniformly or consistently within universities. For example, at the same organizational level as the School of Public Health at Harvard is the Faculty of Arts and Sciences which runs both Harvard College and the Graduate School of Arts and Sciences. My current university uses the concept of an academic "unit". Many units are "schools" (e.g., the School of Public Health or the Information School) but others are not (e.g., the College of Arts of Sciences). There is also an organization within the university called the Graduate School which cuts across and oversees work in different units and schools.
Sometimes, smaller "schools" will not contain separate departments. Sometimes, a large department in a large school might be larger than entire schools elsewhere in the university! Sometimes, the terms might be used interchangeably. In many other cases, they won't. In some cases, one or both terms might not even be used at all!
In line with Benjamin Mako Hill's answer there may be no uniformity between different institutions. In general the following hierarchical structure is applied to universities in Ireland.
- University Name
- College Name
- School Name
- Department Name
So in my case in my university as a History Student it would be the Department of History in the School of Humanities at the College of Arts, Social Science at my university.
In this scenario the School is made up of a number of Departments of specialised study and the college is made up of the schools that would fall under its remit.