A public body works for the government but is not necesarily a part of the government. An institution is any group that is devoted to a particular cause or program, such as college. A school is an institution. In terms of sociology an institution is a pattern of accepted behavior, so a family can be called an institution, but not a public body. Public body is a term that is exclusive to a group that works for the government but is not necesarily a part of the government as stated previously. Hope that helps!

In most people's opinion, they are the same. They are both organizations founded for a particular purpose. A public body is simply a large group of people gathering for a certain cause. This is most often used in politics to represent a number of people. An institution can be an organization founded for a cause as well, like the Institute of Art or the Discovery Institute.