This depends on how you define the question, as the Unification era (1850-1871ish) came around 70-80 years after the establishment of Prussia's military orientated culture under Fredrick the Great.

The key text people usually refer to was written by Carl Philipp Gottfried von Clausewitz called On War, which was not finished when he died in 1831. This book essentially became the text book for most European militiary tactics in the 19th century, and helped provide the philosophical backbone for Prussia's professionalism within its military. However, it is important to understand that the reason Prussia is viewed as a militaristic state, as opposed to Britain who was seen as a naval power, was because culturally Prussia embraced a citizen soldier model for its military under Fredrick the Great in the 1740's, which helped him both win the 7 years war and embed a military class within Prussian society that remained up until the end of the Second World War.

German unification under Bismark, Von Roon and Von Moltke was about as much using the Prussian military as a scapel alongside Prussian diplomatic manourvering. Bismark understood that Prussia could not win a protracted war against any of its neighbours, and everything he did from the 1850's to unification in 1871 was to balance out Prussia's strengths and weaknesses to ensure that Prussia came out on top. Bismark was probably the canniest politician of the 19th century, and while technolicial innovation, such as the Krupp C64 field gun added significant weight to the Prussian military, it was Bismark's behind the scenes diplomacy that ensured that Prussia only ever had to take on one major power at a time. I would argue that this was the major defining factor in Prussia's military ethos, the wedding of military genius embodied in Von Roon and Von Moltke and the diplomatic savaunt that was Bismark. No other European land power possessed this duoploy, and it is arguable that when Bismark retired the newly unified Germany lost its key edge.

I appreciate that you are looking for a more militaristic answer weighted on the Prussian army, and culturally Prussian society was more ameanable to the idea of citizen soldiers and Prussian junkers emeshed within the fabric of power, but personally I would suggest that what set Prussia apart in post 1848 Europe after the revolutionary crisises that rocked most of the continent was the foresight to see that libralism was here to stay and that the state should harness the poltical forces for its own ends. Every action Bismark and the Prussian monarchy took in the 1850-1871 period was to coral those libral forces in service of the state to ensure that both reactionary and revolutionary forces within Prussia fell in behind the greater needs of the state. It also helped that Prussia did not take on any military opponents that it could not manage on its own; it is arguable that in 1870 Prussia could have defeated France alone, but Bismark engineered the whole campaign to bring in Bavaria, Saxony, and the other German states as a defensive collation, which in turn enabled the proclamation of the Second Deutschen Reich in 1871 after the defeat of the French. Within France declaring war on Prussia in 1870 the defensive pact would have annulled as Bavaria in particular was not keen on the long-term implications. Bismark manouvered the whole diplomatic package to both coral those states and ensure France declared war, while Von Moltke and Von Roon ensured that the Prussian and Germanic states had the best trained and equipped military in Europe.

So, to succinctly answer your second question, I would argue that the key difference for unification Prussia was seeing war as an extension of diplomacy and the willingness to wield the army as a scaple in short, sharp winnable wars. This contrasts to France, Austria, and Russia who were either beholden to their monarchs whims or were not economically in a position to wage effective short term wars post-1848. France and Austria both lost to Prussia, France because her military leadership was muddled and tactically confused, and Austria because her military had not evolved post 1848 due to social and economic reasons.

I have posted this answer elsewhere, and it only truly addresses the 18th century process of why Prussian military ethos was different from other Western European states.

1. Between 1713 and 1740, Frederick William I ("the soldier king") affected that he was a maniac with an obsession for tall soldiers in order to hide the fact that his state was amassing a considerable amount of wealth, and a relatively large army (83,000 men). During this time, despite a few small conflicts, Prussia was relatively peaceful, allowing Frederick William I to build up his forces in relative peace. The military consumed 3/4ths of the budget, but that was fairly normal for this time. More exceptionally, Prussia possessed the 4th largest army in Europe in 1739, but ranked something like 20th in terms of wealth and population.
2. In 1732, Prussia adopted a form of conscription called the canton system. Prussian soldiers, as opposed to being composed only of landless men who joined the military voluntary, began to represent more of a cross-section of Prussian society, with many second sons from rural peasant families being conscripted into the military. These soldiers were what we today think of as reservists: they were called up for 2 to 3 months of the year in order to train with the army, and spent the rest of the year at home pursuing work in the civilian world. Non-cantonists, although they were not sent out into the countryside to work, spent 4 days of the week working in civilian trades, as part time laborers. As a result, the military grew, and the economy gained the benefit of numerous workers which would otherwise have been lost to the military. Although Prussian peasants were initially horrified by the canton system, desertion in the peacetime Prussian army was very low, perhaps 7-8%.
3. The Prussian infantry were drilled effectively, and while training using cadenced marching did occur, the Prussians focused on training soldiers to be able to load and fire quickly, what they believed was most important in combat. According to some accounts, the Prussians were able to fire five shots in the time it took most armies to fire two. The exact number of shots per minute is not important here, what matters is that the Prussians had a large comparative advantage
4. In 1740, Frederick William I died, and his son, Frederick II ("the Great") assumed the throne. He inherited a great deal of money and a large and competent military which was largely composed of soldiers who actually wanted to be in his military (cantonists/foreign volunteers). He used this money and army to attack the neighboring territory of Austria, taking a very rich province called Silesia (rich in both population, tax base, and mineral deposits). The Austrian Empress, a woman named Maria Theresa, was unable to get the international support to retake this province as a result of other states declaring war on her around the same time, each hoping to take a piece of her territory. This produced a general war, which some scholars argue was even a "world war": The War of Austrian Succession. As a result of his willingness to negotiate and abandon overall alliances, Frederick II was able to end his war with Maria Theresa in 1745 after winning significant victories over the Austrian Army, which was not as tactically prepared to fight as Prussia. Indeed, the Prussian Army was undefeated on the battlefield during this war.
5. Frederick II knew that Maria Theresa would desire revenge, and Austrian Chancellor Kaunitz was eventually able to convince France, Russia, and Sweden to ally with Austria against Prussia, Frederick realized something was about to happen, and inadvertently triggered what was then a defensive alliance when he attack the neutral state of Saxony in 1756. This precipitated the Seven Years War. In this long and bloody conflict, Prussia was heavily assisted by English money, and English victories over France in both the war in Europe and a wider global conflict. Despite this, and some spectacular tactical victories (Rossbach and Leuthen) earlier in the war, Prussia was nearly overwhelmed. Frederick's preferred tactic, to attack the enemy flank, was quite recognizable by the middle stages of this war, and he only seriously began to change this preferred method by 1760. In 1761, Prussia, despite the resources of the canton system and British help, was out of men, money, and time. Fortunately for Frederick, the death of the Russian Empress caused the collapse of the military coalition facing Prussia, and in the year 1762, Prussia was able to score significant victories over the Austrians. The end of the Seven Years War allowed Prussia a resting period (really between 1762-1792) and as a result, Prussia was able to recover from the Seven Years War, and maintain its position as a military power.

For more reading on this process, I recommend:

Tim Blanning, Frederick the Great, King of Prussia

Otto Buesch, Military System and Social Life in Old Regime Prussia

Christopher Duffy, The Army of Frederick the Great (2nd edition 1996)

-------------------------, By Force of Arms: The Austrian Army in the Seven Years War

John Gagilardo, Germany under the Old Regime,