Everytime when the discussion comes to Prussians, people will always mention the Prussian military tradition. But what does it mean? It's obvious that they were rather capable soldiers for being able to maintain quite a sizeable empire, but was there actually something unique about their military? Something that makes them different from their counterparts both in the past and today?
The most basic element to understand here is that the Prussian state, which evolved out of the older Electorate in the Margravate of Brandenberg, was a very particular system of statebuilding. It was a deliberate artificial creation of kings who chose to gear the state as a support basis for the army, and made deliberate reforms to bring this about. Relative to its size the Prussian army was enormous, and so were its demands on the state. Not every Prussian was a goose-stepping peasant in a uniform saluting Junkers in Pickelhaubes with monocles, because no state can attain an abstract ideal perfection. Prussia came far closer to it relative to its size than others, which meant that it went from 'road stomped over and burned by real powers with real armies' to the most powerful military force in Germany.
The Prussian army being this overmighty was a fragile reed, however. A major defeat like Kunersdorf or Auerstadt would literally bring down the entire state with it as it had nothing for itself except the army. That gave the army a superficial pretense of being apolitical and in reality being the mightiest force in Prussian politics.
The Prussian Army reached its apex under the combination of Moltke and Roon, who in a unique pattern for the kingdom not emulated by their successors in the German Empire of World War I or Nazi Germany in WWII understood both logistics and command, control, and co-ordination were the decisive elements in wars and a state that geared the most direct advantages in these would be able to achieve smashing results while other states would barely have started their mobilization processes. This is what won them the wars with Austria and France, but it was a secret misunderstood by the elite of the unified Germany, and something that politically they couldn't have followed even if they wanted to when they neglected the Reinsurance Treaty and created the alliance system that gave them no military answer.
The Prussians were mighty forces at a military level at the simplest because under the Hohenzollerns they developed an army that had far more manpower in a shorter and medium term than any foe it would face, and that had both commanders and ultimately a general staff that let them develop an adeptness at tactics that no army facing them could meet in a short or medium term. As the Seven Years' War example showed, attrition worked very well against them to a point that another two years for Empress Elizabeth would have seen the breakup of Prussia. And as Auerstadt showed, an actually adept veteran enemy facing them could beat three quarters of their army with a single corps, in the right circumstances.
It's a boring but practical answer of the kind that actually wins wars, and misunderstood and exaggerated by their enemies, who had societies that were unable to gear themselves to be 100% support bases for the military machine, unwilling to pay the price, and too ornery with too powerful a set of local political combinations to achieve it.
I'd recommend Isabell Hull's Absolute Destruction as a look for some of the real legacies of the Prussian military to the German Empire, and how much damage that actually did to the military brain trust and political culture of the unified German state. Particularly since Waldersee and Schlieffen eschewed the deeper lessons of Moltke and Roon for things that were grounded more in political wishful thinking than the kind of hard strategic realism that Frederick the Great and Moltke did well, within limits (and even Frederick gutted a quarter of his army at Kunersdorf when his realism antagonized enough people he barely avoided a total loss and the end of Prussia in the process).
Sources:
https://www.amazon.com/Iron-Kingdom-Downfall-Prussia-1600-1947/dp/0674031962
https://www.amazon.com/SOLDIER-KINGS-House-Hohenzollern/dp/3779680246
https://www.amazon.com/Franco-Prussian-War-German-Invasion-1870-1871/dp/0415266718
I'm sorry if this is against the rules, but I find this essay incredibly relevant to the question from the German point of view, as it was specifically about the "German way of war" from the Prussian time. Fellow Redditor u/Tyrfinn at r/bundeswehr (German army) did refer his essay at r/WarCollege when I was asking specifically about German military culture. He went on to explain the military culture that shaped Prussia and eventually today's Bundeswehr.
To quote his essay:
Greetings folks,
today I want to share with you something I thought about as common knowledge but which I realized has to be explained more often then not. Especially in a world in which everybody knows of Clausewitz but only few of Jomini and their relationship and history.
Robert M. Citino actually argues that by the geopolitical situation of Prussia and later Germany the "German way of war" is one of aggressive and short wars ("Short and lively"). While the result of Prussias and later Germanies geopolitical situations are these "quests for decisive victories" it is in no way a "German way of war" and it overlooks very important caesuras within the German military culture within that time-frame. France during the coalition wars had a similiar geopolitical situation and also relied on aggressive campaigns against their opponents while not having the strength (for a multitude of reasons) to find a decisive victory, did surely not followed a "German way of war". Their situation simply demanded quick and aggressive actions. He is thus looking at symptoms not on a fundamental difference, which I want to explain in this post because until today these differences throw a deep and black shadow.
Do not get me wrong on this point: Citino is an awesome historian especially on the 2nd world war but he was raised in a country with a fundamentally different military culture and tries to apply the principles best known to him to armed forces of a different culture. Allow me a cheap analogy on why this is a problem: As a US citizen you can argue that hamburgers are the best meal a person can eat. A Japanese will argue Sushi is the best meal a person can eat. Give a US citizen Sushi and he might argue that the fish should be replaced by beef and he might even have a point but has he then learned about the limitations and possibilities of Sushi? No. He applied his known principles to a foreign framework and the conclusion he can draw out of it will always only allow him to enhance his own framework but never to understand, compare or even adopt the foreign framework.I also want to point out that, while I'm conviced of the German military culture, I'm not saying it is without flaws and it certainly has its limitations. It has its superiority and its inferiority and especially the 2nd WorldWar shows the shortcomings of its approach.
So, where to start?
We start at the biggest caesura in German military history: The battle of Jena-Auerstedt in 1806. In this battle the Prussian army suffered a decisive defeat against Napoleon. Especially in the battle of Auerstedt the loss of officers caused a complete collapse of command. Losing this war meant for Prussia to lose huge chunks of territory and being mere 50 years after the seven years war reduced to a 2nd-rate power in Europe. What was needed was a reformation of the state and the military. The 5 most famous military reformers of their time were Gneisenau, Scharnhorst, von Boyen, Grolman and Clausewitz. Despite all of their reforms, as e.g. a modern conscription system, the foundation for a modern general-staff system and others, we want to look at the framework laid down by Clausewitz.To understand Clausewitz one has to talk about his "biggest rival" Antoine-Henri Jomini and the different views on war. Jomini understood war as a science and always falls back in "Art of War" on military science and the importance to gather perfect information. Where information is riddled the evaluation of these information is what differentiates an "ordinary general" from a "genius" (see the chapter "Of Reconnoissances and other Means of gaining correct information of the movements of the enemy"). Clausewitz contrary to this view argues that a big part of information obtained is contradictory, an even bigger part wrong and the biggest being uncertain (see chapter 6, "Nachrichten im Kriege"). Further in chapter 7 "Friction" he even states that while in theory the war is simple the amount of different parts that engage with eachother will always cause a loss on information and controllability.
Where they thus differentiate is in the idea how to deal with the fog of war. Jomini tries to overcome the fog of war, Clausewitz accepts the fog of war.
Now, how have both ideas spread around the globe? Surely, since Clausewitz is so well known today he must have been taught more frequently? Unfortunately this isn't true.
Clausewitz was an introvert and humble man who served most of his short live within the Prussian military and who died in 1831. In contrary Jomini was an extrovert, partly bragging person, who served mainly in France and Russia and lived until 1869. His writings were taught in West-Point and over all military acadamies in the world. Clausewitz was known too but never reached the widespread acceptance in scholarship of the 19th century that Jomini enjoyed, except in Prussia in which Clausewitz ideas were the foundation.
While pretty much the rest of the world thus started to centralize commands after Jominian lines, Prussia and Germany had to find a way to deal with the fog of war. Here another famous character appeared: Moltke der Ältere. If information in war can't be trusted and might be contradictory, decisions have to be made as close to the action as possible to be able to react in time to the shifting situations on the battlefield. But how to organize such chaos? The intent of the higher command becomes decisive. Not longer shall a single genius leading a battle he can't overlook anymore but independent leaders should strive towards a common goal set up by the higher command. Auftragstaktik was born.
From the second Schleswig War (1861) up to the first world war there was a huge debate. It wasn't instantly installed on all levels but it gained more and more momentum until during the first-world war not only officers enjoyed the priveleges of independent leadership but also NCOs and the whole army was trained along these lines.
But Auftragstaktik is not a way in which orders are issued, it is a whole culture. For it to succeed, one needs independent, strong characters, who are able to think creatively, who trust their subordinates and superiors and who are allowed to make mistakes. To build the trust of subordinates one has to share their burden, one has to lead from the front, not only to build trust but to make decisions based on own impressions of situations. Thus an army that tries to break the character of young recruits to form warriors will never get the personelle it needs to perform Auftragstaktik. They need soldiers with empathy. This is why the tone within the German armies from the first world war on was far more civil than in other armed forces of their time. This doesn't mean there weren't hardships and drills in barracks and harsh words, but e.g. officers eating with their men became more common, teaching soldiering not breaking and making is the idea of training. After the 2nd world war the Bundeswehr even developped this way of leadership further within the idea of "Innere Führung" which until today doesn't even have a true translation into English because the direct translation "Inner Leading" isn't understandable. So even in Basic-Training the difference in the ideas can be observed.
The whole difference between the understandings of the German way of war and the Jominian way of war was already known to the world before the first world war. It wasn't a secret at all. For example in the 1920s George Marshall was being appointed to the Infantry School in Fort Benning and taught along the same ideas which proved vital for the US in the 2nd world war but as soon as he left the Infantry School everything went back to normal.
Auftragstaktik is not the only point that differentiates the German culture of war from others, but it is the most abstract, most fundamental part that is even today wrongly understood and has far more implications in other areas of military life than just in planning operations and training. Other points that were as vital to the development of the shift are for example the idea of "Citizen in Uniform", the influence of conscription on the society and military, the different ways in which staffs work,...
Also by "neglecting" the "scientific" side of war like logistics and the fact that the German military had for political reasons up to 1945 (and partly to today) a strict apolitical stance, grandstrategy was underdevelopped.