What are the structural differences between the nobility of, say the 18th century, where they were legitimized by a pseudo authority like the Catholic Church, and the so called one percent, where they are legitimized by a pseudo authority like democracy governed by the rule of law.
In other words, legitimizing through "divine right" seems pretty absurd now. Whatever the shenanigans between the nobility and the Catholic Church to work that deal out certainly wouldn't pass the smell test today for most people.
But legitimizing through "democracy governed by the rule of law" smells fishy too - the shenanigans between the so called one percent and the rule of law (i.e. the golden rule: those with the gold make the rules).
Are there real structural differences or is this just an abstraction of the same old same old?
There are substantial differences between the very rich today and in the 18th century. I'm not sure about calling democracy a pseudo authority and shenanigans, but that strays into modern politics which is outside this sub's remit. I'll just talk about the differences in wealth, focusing on the top 1%, between the 18-19th centuries and today.
What they own
In 18th century Britain and France, two-thirds of all national wealth was agricultural land. Today, agricultural land is less than 2% of national wealth. In its place, housing and other capital (including office buildings, machinery, computers, infrastructure) have both become the dominant forms of capital, with wealth split about 60-40 between housing and other capital. Industrial capital is usually owned indirectly - people own shares in companies that own this capital, they don't own the capital directly. But the fundamental change is that the rich in the 18th century primarily owned land, while the rich today own industrial assets (or financial assets such as shares) and housing.
How much they own
Capital is substantially less concentrated today than it was in the 18-19th centuries. Then the top 1% owned about 50-60% of the country's total wealth; today they own about 20%. This is largely due to the growth of a middle class that owns more of national wealth. It has always been the case that the bottom 50% of the wealth distribution owns practically nothing (though they might have quite high income in some cases - a newly qualified doctor will have massive student debt and so negative wealth, but will have quite a lot of income and a high standard of living). However, in the 18th and 19th centuries few people outside the top 10% had substantial wealth (the top 10% owned 80-90% of all national wealth), whereas today wealth ownership goes down to the top 50%. But the very rich today have a much less dominant share of national wealth than they did in the 18-19th centuries.
How they got it
Data on this does not go back as far, but there's been a substantial evolution even since the early 20th century, which you can extrapolate from to get some idea of what things might have been like in the 18th and 19th centuries. The top 1% of income earners in the US in 1929, on average, made 50% of their income from owning capital (i.e. rents on housing or land, dividends from shares, etc.) and 30% from labour (wages for work). (The remainder is mixed income - basically the income made by self-employed people, which is partly pay for their work and partly profit they get from owning their business, so is a combination of labour and capital income.) Today, this has been almost exactly reversed - 50% of the income of the top 1% comes from labour, and 30% from capital. In 1929, the whole of the top 1% earned more from capital than from labour on average, while today this is only true for the top 0.1%. The rich in the 18th century, by and large, got most of their income by passively receiving rents from the capital they owned; the rich today are far more likely to get their income by being paid a very high salary for work they do.
The rich in the 18th century were also far more likely to have inherited wealth rather than earned it, as compared to the rich today. In 1790s France, the top 1% of inheritors could get an income from their inheritance equal to about 25-30 times the income of the lower classes; the top 1% of wage earners could only get an income about 10 times that of the lower classes. Today, the income you can get from inheritance has fallen to about the same amount as that you can get from wages. So in the 18th century the top 1% was dominated by those who had inherited their wealth, while today the split is roughly even. In the US, inheritance has always played a smaller role than in Europe, but the data from the US isn't clear enough to tell. Despite the fall in importance of inheritance for the top 1%, the total flow of inherited wealth today is about the same as it was in the 18th-19th centuries; the difference is that today the top 1% inherit less, but the next 10-20% inherit more than they did in the past. This is linked to the broader distribution of wealth discussed in the previous section.
(n.b. these two paragraphs seem to be saying similar things, but there is a distinction. Capital income as described in the first paragraph may come from an inheritance, but it's also possible that people bought their capital with money they previously earned from labour. The second paragraph focuses only on inheritance.)
Source: Capital in the Twenty-First Century, by Thomas Piketty; specifically chapter 3 for the first section, chapter 10 for the second, and chapters 8 and 11 for the third.