The trouble with a monarchy is threefold - there is the inevitable corruption, the high chance of incompetent rulers (heh, not that democracy would be immune to this) and the question of class society. In theory, monarchy might be a good way to govern just like a benevolent dictatorship is, with a leader that has been grown from early childhood to become a good head of state. In practice, problems tend to mount and grow on each other untill they detonate in a civil war between the nobility, a coup or a popular uprising, much like in dictatorships.
Let's take a look at the Roman Empire, which was not a typical monarchy in the European sense of the word, but close enough. When there were good..no, not "good" but competent and ruthless emperors like Augustus and the five good emperors, things went pretty fine. But when incompetent maniacs had the throne, things detoriarated fast. My personal opinion is that Commodus was one of the main reasons for the destruction of the Empire, as he actively encouraged corruption and sold many important government posts to his completely incompetent friends, as well as ruining Roman economy for decades to come. Charlemagne is another good example. He almost started Renaissance centuries "too early", but pretty much all was lost by the squabbling of his (grand)children, especially the cultural advances.
The third question, that of an inequal society, is of course relevant only if one finds that to be undesirable or unethical for some reason. That is what monarchies lead in to, unarguably - at their heart is the idea of rulership passed from parent to a child, which will inevitably form an idea of some bloodlines being more worthy than others. It doesn't matter if it is based on a divine will or something else in its arguments, the end result is a system of privileges and special rules. This in turn highly increases the chances of corruption and incompetency at high places. Of course, if we take this in to a science fiction setting, this special caste of people might be genetically engineered to really be superior to anyone else. They might even be completely different species than their subjects..but that takes the discussion to another direction.
One good question to ask to illustrate my point is - are there any kind of special ills and mistfortunes that are only possible in a monarchy or a similar authoritarian system? On the other hand, are there any special good, useful things that are only possible in a monarchy? (When I say "monarchy" in this post, I mean a real, authoritarian government like feudal or absolute monarchy or some variation of them, not a system where the monarch is a glorified symbol of the nation with little real power.) Just look at Saudi Arabia for a real, modern monarchy and how well it is working for everyone... Benevolent dictatorship is a nice idea, but not very likely to work to anyones advantage but the dictators. Unless of course you want to make ME the Absolute Emperor of the Universe, I promise everything will become just wonderful then.
What comes to progress and governments, elitistic governmental forms with rigid social structure tend to stagnate. The upper classes have what they want and need - they are all for status quo so that they can keep being the top dogs. If they want something, it is to hit someone else on the head with it. If there is no competition, there is no need for advancement. A bit like how the Chinese hit a technological and cultural plateau that lasted for a long time. The Roman Empire had the same problem towards the end of its reign. I think the nature of the Third Imperium as a class society where the nobility and the megacorporations have the power and the needs to defend their interests extremely efficiently is one good reason why technology has not been advancing significantly for some time. Nobles don't want anything that would upset social balance while the megacorporations want to keep the markets stable. Thus together with the trouble of a huge scale (thousands of member worlds!) political inertia and economical pressure are strongly against genuine innovation.