9I agree, it is a very interesting POD. However since the men died at a young age at 25 and just three years in power it give way to many speculations and scenario's.
He was married with Mary Stuart thus direct lineage with he English King. As a high Noble he had enormous ambitions and likewise a spending drift as a real king.
I do have a few idea's.
Short term; alliance with France against Spain, possible ending in a same alliance with the Commonwealth even Cromwell was the murderer of his father in law.
(Dynastic schemes in Europe are ruthless, even it sounds unlike you ally you self with a murderer of the father of your wife)
Re-opening of the war in the Southern Netherlands and over seas.
Making the state finances of the Dutch Republic weak, not sure if this have negative effect on the economic growth of the Republic, it seems there was a economic crisses after the Peace of Westphalia, so it looked if the (land) war was beneficial for the economy. the war at sea probably is negative if the Dutch loses the upper hand.
Increase of the State army and moderate increase improvement of the navies, at least not as ambitious as under Johan de Witt.
Best guess the war will be ending as in OTL in 1659.
William II is more aggressive as is father Frederick Henry, how ever also less tolerant towards Catholics. The Military campaign in the Southern Netherlands would be more a war of conquest and submission than a kind of liberation as under his father. Which can be causing resistance for the future in the South.
William II neutralized the Bicker and the Graaff factions who opposed his father in gaining more power in the Republic, preventing with financial aid to the Hapsburg, the conquest of important cities in the South as Antwerp, Ghent and Oudenaarde.
Even slower as expected I do think William II can manage the conquest of large parts of the Southern Netherlands. Possible the Southern Netherlands will be divided as agreed in 1632 or, depending on the out come of the military campaigns, in a more traditional manner, by the borders of the counties, fiefs and duchies.
I suggest France would get Artois, Chambrai, Luxembourg, while William II would get entire Flanders, Brabant, entire Hainout, Limbourg and Namur.
Possible exchange or increase of trade post in Asia and the Atlantic at the expense of the Spanish. Possible peace treaty with Portugal were the Portuguese will size large parts of Brazil back the the Dutch as they proposed, but rejected by Johan de Witt, who in stead agreed with a large sum of money ( which never came).
William II Orange Nassau would act and behave as a real monarch, how ever the resistance within the Republic to elevate him a a real monarch would be to great.
Long term.
If William II would live longer it is very likely there will be more children. On the other hand it would be possible they will not survive child hood, the same can be siad about his eldest son (OTL) William III which had a weak health and had asthma.
The 17th and early 18th century wars between the Dutch Republic, Dutch war 1672-1678, Nine years war 1688-1697 and the War of the Spanish succession were more or less a consequence of the the be trail of Louis XIV by Johan de Witt. At least that is how Louis XIV viewed the sudden opposition organised by Johan de Witt after Louis XIV smashing success of invading the Spanish Netherlands and Loraine. Leaving Louis XIV with a rather poor result at the peace Aix la Chapelle 1668, compared with the territorial gains his army made in the war.
Since the Southern Netherlands were divided it is most likely Louis XIV would focus on Loraine and other (Spanish) Hapsburg possessions surrounding France rahter than nible parts of the the now larger Dutch Republic which is still an ally.
No doubt there will be tensions raising in the second half o the 17th century, but this will be due to the economic success of the Dutch Republic.The economic success would no doubt be still happening, since the state structure would prevent any absolute power for William II, unless he want to risk a civil war, which ruins his coffers.
Most likely the Dutch Republic would evolve in a kind if symbioses between a wannabe Monarch and a council of Patricians which, even opposed to each other also needed each other.
Centralisation of the Republic could go a bit faster due to this domestic power struggle. Due to the fact that Antwerp and Flanders and Brabant are part of the Republic and will be part of the State General and not be a subdued territory. ( other wise they will revolt in a few years) the power of Holland and Amsterdam would be a bit watered down, whill be in favour of all other Provinces as well.