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 that officially have the definite as an integral part: the Bahamas, the Gambia, and the Congo (the latter found in country names 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.