In daily usage, the terms ‘nation’ and ‘country’ are used interchangeably. Official terms like ‘international match’, ‘United Nations’, ‘Member States’ etc serve to blur the differences even more. But a closer look will reveal the difference between a country and nation.
In broad terms, a country is a group of people governed by a government, which is the final authority over those people. There is a political setup that governs everyone in the country.
A nation refers to a group of people bound together by common language, identity, ethnicity, history etc. Cultural commonality usually leads people to identify themselves as a nation.
When the nation organizes itself under a government that is the ultimate authority, it is also known as a country. The terms ‘country’ and ‘State’ are often interchangeable. The term Nation-State is used as well.
Culturally connected people who identify themselves as a nation can do so while simultaneously being part of a country. Several Native American groups identify as nations; the Cherokee Nation is an example. But they ultimately belong to the State, or country of the USA. Note that upper case ‘S’ is used when referring to the US as a State. Lower case is used for the states that are sub-divisions of a country, like Florida, Ohio etc.
Another case in point is the UK. The country is the United Kingdom, within which it accepts the identities of Scotland, Wales and Northern Ireland as member nations. Which is why you see Scotland playing England, and Wales playing Northern Ireland in international matches. But as a State, the UK is the authority, and there are limits on the powers of the nations.
Some countries do not encourage the idea of nation within its borders. Iraq, for example, doesn’t recognize the Kurdish people as a nation, because it is a nationalism that translates into a move for independence as a State. In several countries, the idea of nationalism is encouraged by governments, to make everyone look upon their country as their nation.
Another related term is ‘territory’. These are regions that operate on their own, but ultimately depend on another country. So Puerto Ricans might identify with Puerto Rico as their nation, which is a territory of the USA, which is the country that ultimately governs Puerto Rico.
Beyond casual usage, the term we use is a function of how we choose to recognize groups of people.