Generally speaking, these words are the same. When we talk about nation we are also speaking about the country. For example, these two sentences provide the same context: "The Nation will vote next week." "The country will vote next week." If you go further into comparative politics, the definition of what 'nation" is can change (but that is beyond the scope of your question).

Christopher has a wonderful explanation. The word COUNTRY has other meanings. It can refer generally to the lands which are outside of a city. One can be said to live "in the country" which means rural areas, not urban or city-like. . Some people write or speak of themselves as being "from the country" while others indicate they are "from the city". .

country = territory nation = people

They mean the same thing usually. The Nation means the country so if we say nation, we are usually talking about the country. But the country can also mean in rural areas, or it is othherwise known as the countryside. An example of which is: "People all over the nation are gathering..." And "People all over the country are gathering..." They have some other symbolic reasons, but they are very in depth and 'beyong the scope of your question'