Sometimes this difference is well known to everybody speaking this or that language. Just some samples for you to understand or remember better.
Russian 'students' do learn, but in colleges and universities only. Russian 'spectacle' is a theatrical performance of a play only. Russian 'sinus' is the mathematical sine only (except for the Russian medics, maybe), etc., etc.
But sometimes it's not the case and diversity of meanings is not that great or evident. And these 'hidden' discrepancies of meanings can be really misleading in communication.
Here, a popular Russian joke:
On the Russian-Latvian border, during passport check a Latvian officer asks a Russian tourist in English:In fact, it's not that fun. In Russian language the word 'оккупация' — occupation — has only a negative connotation, it is often illegal and made by states using military power. Germany and allied troops occupied a huge part of Soviet territory during the WW2. After the WW2 the allied powers occupied Germany. This is the only meaning of the word in Russian, we absolutely cannot have something like "Occupy Wall Street" or an "occupied seat" in a restaurant.
— No-no-no, just traveling!
This is one of the reasons why we Russians (well, most of us) do not accept talks about the Soviet occupation of the Baltic countries in 1939—1940. Formally, all the democratic proceedings required by the intrernational law of the time were done in these countries, and they became Soviet republics, were incorporated in the USSR in a peaceful way. In fact, for an English speaking person they were occupied (albeit in a formally legal way and without any military actions); but for Russian speakers they were not at all, just because the USSR did not conquer these territories using its military power and losing lives of its soldiers.
A similar thing happens with annexation (e.g., of the Crimea). In Russian, 'аннексия' is a violent, 'forced attachment' of some other country's territory. However, in English annexation is "the incorporation of new territory into the domain of a city, country, or state" — neutral definition without any negative connotation. Thus, in Russian using the Russian word 'аннексия' to describe the reincorporation of the Crimea into Russia is kind of a clear sign marking, say, anti-Putin political position of a person.
Unfortunately, plenty of inadequately educated translators and interpreters seldom know and feel these differences. And it's not just a job; it's a hard and very responsible work — to make peoples clearly understand each other across state and language borders.
Hopefully this helps a bit.