A country is a geographical grouping of people united under a common government. A nation is a group of people united by a single culture and set of traditions. It is possible for multiple nations to exist within a country.
Native Americans are a good illustration of a nation existing within a country. Because they were born in a land governed by the United States, Native Americans are American. But they are also members of their respective nations. For example, the Navajo nation is comprised of all of those Native Americans who share the Navajo culture. The Lakota nation includes those native Americans who identify with the Lakota culture. They are both native tribes. Members of both are Americans. They are two separate nations. This is why when citizens of a country are referred to as "nationals" it is a cultural and not a political reference.
This is not to say that the relationship between nation and country is not homogeneous in some countries. In fact, the close relationship between nation and country in most cases is why the two terms are so frequently used interchangeably.
The terms can also be used to show deliberate dissent. When the American nation is referenced in relation to politics, it is often to highlight the disagreement of the majority with government action.