While the Prussian military had performed well in the Seven Years' war, that was the last major (Europe-wide) conflict it had fully committed to; it had left the French Revolutionary Wars earlier than Austria or Russia and still used the doctrines and tactics of Frederick the Great. These emphasised rigid drilling to produce tightly-controlled and precise movements and fast rates of fire, and largely (although not entirely) discounted light infantry and large-scale use of skirmishers. By 1806, the Prussian Army was largely made up of troops who hadn't seen large-scale combat, and those officers -especially those in high command - with serious military experience were the veterans of the Seven Years' War, fifty years before.

The Grande Armée, meanwhile, was a veteran, proven force. Its generals had upwards of a decade of combat experience, and the Revolutionary governments' general policy of rotating generals through different commands to prevent them gaining too much influence with their troops and overthrowing the government meant that they were also able to perform in a variety of situations and locales. The army as a whole was a mix of veterans of the Revolutionary Wars, some of whom had been fighting since 1794, conscripts blooded into soldiers by the victorious Austerlitz campaign, and fresh drafts integrated over the course of 1806.

In terms of doctrine, the Grande Armée had also pioneered several innovations: the corps d'armée system allowed the French to move fast and separately, but also concentrate for battle when needed, since each Corps could march, fight, and forage independently. This extension of the Divisional system hadn't yet reached the Prussians, and neither had French infantry tactics. French doctrine emphasised clouds of skirmishers to reconnoitre and harass enemy units, while a flexible formation at division-level allowed battalions to smoothly move from column to line to square as needed.

The French also had an advantage that the Prussians had lost: unity of command. William Frederick, unlike his grandfather Frederick the Great, was not a military man, and so military command was exercised by a council of generals, who, in the preparations for war with France, were unable to agree on what the plan should be. William Frederick therefore brokered a compromise battle-plan. The French, on the other hand, had absolute unity of command under Napoleon, who, while perhaps not the peerless legendary figure he is often made out to be, was nonetheless a highly competent, intelligent, and hard-working commander. Napoleon's style of warfare emphasised speed of manoeuvre to find, cut off, harry, and ultimately destroy the enemy army, in contrast to many of his opponents who at this point still somewhat focussed on the control of territory and the protection of their lines of supply. Further, the Marshalate was - to some extent - trusted to carry out Napoleon's orders and interpret their objectives in order to destroy the enemy at this point. The fractious Prussian war council was ill-equipped to deal with this style of warfare and unable to respond to ongoing developments with the speed that French command could.

The actual story of Jena-Auerstedt is long and complex, but in many ways the weaknesses of the Prussian army showed themselves in the engagement; the slow movement and fractiousness of Prussian command resulted in the army being split, with half facing Napoleon and the majority of the Grande Armée at Jena, while half the Prussians faced Marshal Davout's III Corps at Auerstedt. The early death of the Prussian commander at Auerstedt resulted in command falling to Frederick William, who hesitated and was unable to commit decisively to one course of action, resulting in III Corps, despite being outnumbered two-to-one and hugely outgunned, withstanding Prussian attacks and actually breaking the Prussians. At Jena, once the Prussians had been broken, Napoleon launched a ruthless cavalry pursuit, ensuring that the Prussians would be unable to regroup. In the following weeks, the French constantly harried the retreating Prussian forces, breaking apart forces and destroying all organisation, effectively dismantling the Prussian army.

This is not to say that everything went right for the French; the only reason Davout and III Corps had to fight alone is that Marshal Bernadotte with I Corps refused to march to III Corps' aid despite being within earshot of a raging battle, and failed to move to support Napoleon either. This almost led to Bernadotte losing his Marshalcy, but his fresh Corps proved valuable to pursue the Prussians. Further, Napoleon badly miscalculated and believed the Prussian force at Jena was the main and only Prussian army. However, the strength of III Corps and of Davout as a commander, and the organisation and doctrine of French forces generally, meant that despite these errors the French defeated Prussia utterly.

In conclusion: while Prussia had the best command and drill in Europe (arguably) in the Seven Years' War, the Prussian Army was largely using those same generals and doctrines fifty years later, while the French had comprehensively changed the nature of warfare with new doctrine and approaches, and were a hardened force forged by ten years of warfare, led by a consummate general who had a highly-organised and well-trained force that could respond quickly to unexpected developments.

Sources:

The Campaigns of Napoleon by David Chandler

Swords Around a Throne by John R. Elting