A hundred years ago, when most countries in the world were not independent, the difference would have been more evident. It just happens that we live in a time when most of the world’s land area is part of one independent country or another, and so we tend to equate “country” with “sovereign state”.
Consider the case of Ireland, for example. Even before it was independent, it was known as a country. To this day, the “country” of Ireland is not coterminous with the territory of the “state” that is the Republic of Ireland. (And just to be clear, that statement does not imply any territorial claim.)
The same could be said 100 years ago of Korea, or Poland, or Egypt, or any of dozens of countries around the world. They did not suddenly become countries at the time of independence.
Scotland is a country in much the same way as Ireland is, despite the fact that Scotland is not a sovereign independent state.
Other non-sovereign places with the legal status of “countries” are Greenland and the Faroe Islands (each a country of the Kingdom of Denmark) and the Netherlands Antilles (until 2010 a country of the Kingdom of the Netherlands).
I want to disagree with Derleth slightly in the assertion that the Isle of Man is not a country and is effectively ruled by the UK. It is correct that the UK is sovereign in the Isle of Man and that the UK looks after foreign affairs and defence matters. But the Isle of Man is self-governing in all other matters and is not subordinate to the UK Parliament.