Suppose a person is raped and we arrest the rapist. Should his status, whether he's a senator, professor or an ordinary man, play a role in the adjudication of the crime and subsequent punishment? I'm betting that the average person would answer that the law against rape is general and non-arbitrary and one's status should have nothing to do with the adjudication and punishment for the crime. That's precisely what is meant by "rule of law." Or, as English jurist A.V. Dicey put it, "Every man, whatever be his rank or condition, is subject to the ordinary law of the realm and amenable to the jurisdiction of the ordinary tribunals."
Law in the true sense consists of a set of general rules applicable to all persons, as opposed to laws that are simply orders by the legislature requiring particular people to do particular things. Rule of law is critical to the preservation of liberty. Unfortunately, most Americans neither understand nor appreciate this, and we are increasingly being ruled by arbitrary orders and privileges based upon one's status. Let's look at a few of them at the national level.
During the 1980s, many savings and loan banks made huge losses because of chicanery, stupidity and unwise investments. Congress bailed them out. In 1987, when the stock market crashed, many Americans incurred large losses because of unwise, perhaps stupid, investments. Equal treatment before the law would require that if Congress bails out one American who makes unwise or stupid investments, it should bail out any American who makes unwise or stupid investments. Instead, Congress gave particular people privileges because of their status.
A rule of law regime would require that we scrap the Internal Revenue Code in its current form. What justification is there for different tax treatment of one American because he has a higher income, minor children or receives his income from capital gains instead of wages? Equal treatment would require Congress to figure out the cost of the constitutionally authorized functions of the federal government, divide it by the adult population and send us each a bill for our share. You say, "What about the ability-to-pay principle of taxation to pay for the cost of government?" That's just a politics of envy concept that would be revealed as utter nonsense if applied to any other cost. Would you apply the ability-to-pay principle to, say, gasoline or food purchases where different prices are charged to different people depending on how many dependents they had, their income, or whether their income was derived from wages, dividends or capital gains?