As you imply, there's more to it than just that it was a multinational state. Many multinational states existed then, and exist now, and did not fall simply because of that reason. That's a very simplistic and, frankly, wrong, explanation of why any state falls apart. It is just a description of its nature, not a cause or effect.

That aside, one reason Austria-Hungary had more problems than the PLC is due to historical context. The rise of nationalism made multinational states, particularly those of non-democractic or authoritarian states, harder to maintain. Elites of such states often did not concede rights and privileges to subjects who were of different ethnicity or religion than the ruling class or people. In the 1840s, Europe went through a major "liberal revolution" where many countries experienced major civil uprisings, generally against authoriarian or absolutist government, and for more rights of workers, which coincided with the rise in national awareness of minority peoples.

Austria-Hungary made some concessions, such as including the Hungarians as a nominally "equal" nation, with the dual-monarchy. Ultimately it did not do enough, especially in the Balkans, to integrate its various ethnic groups in a time where self-determination was a driving factor in politics.

The PLC was consumed by external forces before all of these events and trends. We could talk about the way the integrated (or didn't) various ethnic groups in the Commonwealth, but the big takeaway is that nationalism was not as big a factor in its time as it was in the late 19th and early 20th century.

The degree to which the Polish-Lithuanian commonwealth was "multinational" is somewhat complicated. As usual, it is useful to note that a "nation" as a mass body of ethnically homogeneous citizenry did not exist in the Polish-Lithuanian commonwealth.

There was talk of a "nation", but that "nation" was not Poles or Lithuanians, but was instead the nobility. In that sense, the "nation" was limited to around 8-9% of the population (depending on time period).

Most people, rather than being nationally organized, were organized into different legal communities. Jews would have their autonomous communities and might be given a monopoly over a good or service. Towns would have charters defining town membership and rights. Etc. Peasants, the vast majority of the population, had few legal rights and were organized by the whims of whoever they owed legal obligations to.

My personal favorite story describing the lack of nationalism amongst the peasantry is when, after the third partition in 1846, the peasants in Galicia slaughtered the Polish nationalists attempting to organize them in revolt against the Habsburg Monarchy, because they felt the monarchy was their only defense against the nobility.

So, for the Polish-Lithuanian Commonwealth, the issue is that while there was a great deal of linguistic, religious, and legal pluralism (insofar as different classes had different legal statuses), this diversity was not centered around the nation as a limited, sovereign community, unless one includes the nation of nobles in which case there was no real competing nation.

As for Austria-Hungary, while people talk about its failure due to nationalism. I think that's a bit overly simplified. Austria-Hungary had a number of nationalist elements, be it the Hungary nobility who imagined Hungary as a separate legal nation and therefore demanded a degree of autonomy from Austria or the various Czech and Polish nationalist middle classes, etc. These created pressures for regional autonomy. But they the cause of the fall of Austria-Hungary was more complicated than just nationalism. It was caused by pressures of WWI, the success of nationalists, and the harsh military dictatorship imposed on the people by the state.

EDIT: Just to add re: the Austro-Hungarian state. The state was not as authoritarian as often described. Austria had universal manhood suffrage by 1907. An argument that Judson makes is that it was the various ways that activist had to interact with the state (through municipal governments for example or because of the various crownlands) that made nationalist appeals more popular, because it gave a way for activists to frame their issues in a way that they thought would have cross-sectional (e.g. cross-class) appeal.

Readings:

Poland in the Modern World: Beyond Martyrdom by Brian Porter-Szücs.

Poland: The First Thousand Years by Patrice M. Dabrowski.

The Habsburg Empire: A New History by Pieter Judson. His work is markedly revisionist, going against the older model of Austria-Hungary as being a "prison of nations" but while some argue that nationalizing was happening more quickly (see: Mark Cornwall in his review in Austrian Studies vol. 24) it seems to have a mostly positive impression (see: the Central European History forum volume 50 issue 2).

Nationalism's rise in the late 1800s, early 1900s was the biggest contributor to the fall of the Austro-Hungarian empires fall. With nationalism, the people of the different ethnicities were calling for their own state to be created and recognized like how Austria recognized Hungary as an equal sometime in the mid 1800s. Of course the empire, especially the Hungarian part of it, wasn't about it. The calls for their own equal states within the empire or their own independent state kept growing and growing. In fact, that's the reason Franz Ferdinand and his wife got shot by Serbian nationalists.