The advantages of a stable government are primarily the stability itself. Its economy tends to be more stable, its military, if necessary, more loyal, and its citizenry more secure. The United States, as the world's oldest surviving republic, is testimony to that fact. Its stability is a primary reason that it has the world's largest economy, is the last surviving superpower, and its currency sets the standard for the rest of the world. From a historical standpoint, the stable government of the United Kingdom was a substantial factor in the Industrial Revolution originating there.
In contrast, unstable governments (for which Latin America is notorious) do not have stable economies or currencies, their military is often loyal to a military leader as opposed to the government; and its citizens can never feel truly safe. With unstable governments, there is the constant fear of rebellion and the attendant bloodshed. The safety and welfare of a nation's citizens is quite often dependent on the stability of its government.
It should be noted that oppressive governments, such as dictatorships are by their very nature unstable. They seldom survive the lifetime of a single ruler. North Korea is a possible exception to that rule, but North Korea is perhaps the epitome of an unstable country whose citizens are starving.