What is the difference between Ukraine and the Ukraine?

There is something fundamentally wrong with the statement that “The Ukraine is the way the Russians referred to that part of the country during Soviet times”. Russian has no definite article, and as far as I know, the Russian name for (the) Ukraine has not changed since the country’s independence.

‘The Ukraine’ is how English-speaking people have traditionally referred to the country—since long before the Soviet Union was ever a thing, too (at least as far back as the 17th century).

There isn’t one, specific reason why some countries acquire the in English. In some cases, it’s because the name is semantically recognisable as referring to a specific thing (the United States of America, for example, refers to a particular set of united states); in others, it is either random or due to some historical meaning of the name that is no longer clear (The Gambia, for example).

There is a tendency that countries with plural names (as well as archipelagos) have the definite article: the Netherlands, the Philippines, the Canaries, the Bahamas, etc.

There are only two country names and one ‘area name’ (for lack of a better word) that officially have the definite as an integral part: the Bahamas, the Gambia, and the Congo (the latter used in the names of two countries that make up the ‘area’: Republic of the Congo and Democratic Republic of the Congo). In all other cases, it is a matter of euphony and convention—to many people, it simply sounds nicer to include the definite article. With some countries, the tendency to include the article has diminished, making forms with the article sound quaint and archaic. Who these days says ‘the Argentine’, for instance, rather than just ‘Argentina’?

The article does not, however, in and of itself indicate that the country whose name it is attached to is seen as a part/colony/state/subject of another country. That interpretation is—to my knowledge—only applied to (the) Ukraine, and it was invented by the Ukrainians themselves.

There is nothing wrong with avoiding the article and thereby pleasing those Ukrainians who feel that it somehow belittles them—unfounded and unwarranted as such a feeling may be—but outside of official statements where a specific guideline dictates what form to use, there is also nothing wrong with using the traditional English name, with the article.

In brief: the only difference between ‘Ukraine’ and ‘the Ukraine’ is political correctness and official guidelines.

I don't believe that this is a purely English-language distinction, as there is a direct analog of this pattern in Russian. To say the phrase "in Ukraine" in Russian, you can say "в Украине" or "на Украине". "В Украине" literally means "in Ukraine", whereas "на Украине" literally means "on Ukraine". "Нa" is usually used for regions or geographic features, but not generally for countries, and indeed when Ukraine was part of the USSR, it was more common to say "на Украине" than "в Украине". When Ukraine became independent, the phrase "в Украине" was encouraged to emphasize the country's independence. My Russian teacher made it clear that it was polite to say "в Украине" for that reason.

Alaksiej's etymological answer (which is supported by Etymonline) seems to be consistent with this pattern. "On the outskirts" in Russian is "на окраине", which is similar enough that I'd take it as the source of the "на" pattern.

Etymonline "Ukraine" entry

Some additional evidence in response to comments: this Russian-corpus ngram history shows how quickly and recently "в Украине" entered the lexicon.

Ukraine means literally on the outskirts (and that was true from the Russian Empire point of view). I guess the would denote it's a descriptive name rather than a country name (that outskirts, not the other ones). So removing the makes sense, since currently the state is independent, its name is unique and doesn't require any additional classifiers.

"The Ukraine" is a region of the world that has existed for some time. "Ukraine" is the name of the country created after the fall of the USSR that more or less governs this region. Parts of what was usually considered part of the Ukraine might be in neighboring nations.

Similarly, "The Great Plains" is a region in North America. It extends across part of the US and might extend into Canada, depending on how generously you draw the boundaries.