The defining trait of an "oligarch" (in the current usage of the word, not the ancient Greek) is someone who got the power and money through a combination of crime and state power, which are usually not even separated.
No, it's not just a difference between "a rich guy from Russia" vs "a rich guy from USA". The rich guy in Russia most likely had his competitors killed, literally, and the police turned a blind eye. Denying this is ignoring how the post-communist countries work.
And yes, sometimes people get killed in USA, too. But compared with that, in Russia they get killed for trivial stuff. As an example, imagine that Russia would have its equivalents of Bill Gates and Linus Torvalds. What would happen is that the Russian-Gates would bribe the police to put a bullet in Russian-Linus's head, most likely on a busy street in daylight, to send a message to other wannabe troublemakers. This is how business is done in the post-communist countries.
You can complain about oligarchy in USA, and you can make a few good points, but there is still the difference that in USA you can get where Bill Gates is without paying to have your competitors killed. So it is not the same. Therefore it makes sense to use different words.