They’re actually very similar - in both cases, government power is effectively concetrated in a small group, but for different reasons and levels of formality.
Plutocracy: This is where government power is held by the wealthy. However, it seems that this is informal – the wealthy are powerful simply because they have wealth and therefore influence. The government will do things to make them happy, but they hold no official government position.
Oligarchy: This is where government power is formally held by a small group of people, but for reasons other than wealth. Often it’s religion – the Muslim clerics of Iran, for example. Another example might be royal blood, or simply how the government is designed.
You often hear “oligarch” in the context of the phrase “Russian oligarch,” but now I wonder if that’s correct. Should it be “Russian plutocrat” instead?
There’s actually full a Wikipedia page for “Russian oligarch,” and it says:
Russian oligarchs are business oligarchs of the former Soviet republics who rapidly accumulated wealth in the 1990s via the Russian privatisation that followed the dissolution of the Soviet Union. The failing Soviet state left the ownership of state assets contested, which allowed for informal deals with former USSR officials (mostly in Russia and Ukraine) as a means to acquire state property.
“Business oligarchs” links to another page, which says:
A business oligarch is generally a business magnate who controls sufficient resources to influence national politics.
So, this makes me think that “plutocrat” is more applicable, but perhaps Wikipedia is differentiating between “business power” and “wealth.” Nowhere on that page does it discuss wealth, just the usage of monopolistic tactics to amass power to influence politics. Wealth and business power just highly correlated.