In both a plutocracy and an oligarchy a relatively very small group of people wields all the power; majority rule, if it exists, occurs only in token form. Furthermore, both governmental systems do not require a parliament nor a constitution, although these are not obstacles either. There are, however, a few nuanced differences between the two governmental systems:
Oligarchy is the system in which a nation is governed by a few powerful people. The basis of this power is unspecified but sticky; it can be passed on by means of elections as well as inheritance.
“A ruling group is a ruling group so long as it can nominate its successors.” ― George Orwell
Plutocracy is the government of the wealthy, who are powerful because of their wealth. As opposed to oligarchies, plutocracies usually enjoy elective successions – in one way or another.
‘Reagan’s story of freedom superficially alludes to the Founding Fathers, but its substance comes from the Gilded Age, devised by apologists for the robber barons. It is posed abstractly as the freedom of the individual from government control — a Jeffersonian ideal at the roots of our Bill of Rights, to be sure. But what it meant in politics a century later, and still means today, is the freedom to accumulate wealth without social or democratic responsibilities and license to buy the political system right out from everyone else.’ ― Bill Moyers, in his “For America’s Sake” speech (12 December 2006), as quoted in Moyers on Democracy (2008), p. 17