I have always understood the expression "to speak the Queen's English" to mean that one is attempting to speak in manner above his or her social station.  The Best English is the Queen's English and all other dialects and variations are lesser versions of that.  In England, the social stratifications were very delineated and the people of one class could usually identify other classes by they way they spoke.  The more formal and proper and educated the diction and syntax, the higher the class.  It was typical of people to try to speak one step above their own, so as to appear higher on the social ladder.  The nobility then were most attempting to speak the King's/Queen's English.  I think that this attempt to use language to one's benefit is still alive and well today.  We don't write with slang in our formal work, but our Facebook page is a different story.

I would have to say the strict adherence to the ideas and the mere presence of the Fowler's book "The King's English" is a bit outdated by modern standards.  Part of this might be due to the nature of languages being something that represents life and vitality.  Languages change over time and the notion of being able to specifically pinpoint and enforce how a language is meant to be spoken is antithetical to this idea.  Additionally, I think that the outdated nature of the concept of "the King's English" or "the Queen's English" is enhanced by the globalization and technology that have helped to make English such a universally accepted language.  Given how so many text or use technology as a means of communication, there is little to legislate this use.  For example, when someones texts "C U L8tr,"  and uses it as part of their own lexicon, it shows that the manipulation of language has become common understanding and use as to the meaning of such a phrase. While it might cause the Fowlers to rotate at a very fast rate in their graves to see such a use of English (which might not even fit their standard conception of the language), it is something that has become a frequent and common part of the language.

What is King's or Queen's English, and how has it developed and remained important?

King's English or Queen's English is simply the accent of the standard English language as used in England. It is also commonly referred to as Received Pronunciation. You sometimes may see it specifically defined as the accent of standard English used in the south of England, but native speakers throughout the country use it.

The definition of the accent refers specifically to the south of England because it is generally thought that the language rose from residents of southern England before spreading to London and mixing with speech elements from other parts of the country.

It has often been revered in England as the accent of wealthy, powerful, and important people. It actually became associated with spoiled rich kids - people who were born into priviliege but didn't do anything to earn their status. King's/Queen's English started carrying a negative connotation with a lot of people and schools across the country even became more relaxed with formal education, allowing the use of other regional accents.

What are the King's and Queen's English, and why are they important today?

To me, the importance of this sort of Standard English is that it serves as something of a lingua franca for people around the world.

There are, of course, many variations on English since it is spoken in so many countries.  However, there is generally a "correct" English that is taught that is based on the "Queen's English."  Because this standard version has come to be taught worldwide, it is very important today.