Let's say that somehow, enough Neanderthals survived after the last ice age to be able to form a centralized government, economy, etc. Would this even be possible? What events could lead to this?
They Pretty Much Already Did
Neanderthals are now known to have interbred extensively with human beings, to the point that a non-insignificant portion of the modern human genome (5% if I recall right, depending on population?) is from Neanderthals. It was previously thought that sub-Saharan African populations almost exclusively lack Neanderthal genes, but it has since turned out that there was a significant back-migration into Africa throughout history (because surprise, humans don't migrate in one direction) and everyone previously just assumed groups like the Yoruba and Khoisan had no Eurasian ancestry. It turns out they have Neanderthal genes too.
Neanderthal populations are in general thought to have been much smaller than Homo sapiens due to the limited amount of geographic area in Europe/western Asia south of the glaciers, and the cold tundra/steppe climate in most of Europe during the last glacial maximum would have restricted population sizes. Africa was much more hospitable and could support larger populations. When the glaciers melted Neanderthals would have been swamped by Homo sapiens coming from the north regardless of how well they weathered the ice age (and they didn't seem to weather it well), and in many cases it seems the larger immigrating Homo sapiens population may have just absorbed the smaller Neanderthal ones (there's also evidence of violence, but fossil evidence suggests a complex interplay of absorption, interbreeding, and warfare. You know, like humans usually do).
As a result, any predominantly Neanderthal civilization (lets say they didn't get screwed over as hard by the ice age and survived in large numbers) isn't really going to be Neanderthal as we would define it, but Neanderthal-Homo sapiens hybrids. In fact, places like Africa and Asia would likely have more Neanderthal ancestry than the present day, because you would have Neanderthal populations migrating around with the advent of boats and horseback riding and hence more opportunities for intermarriage. The same goes for Denisovans. Whether this results in behavioral differences or whether this merely results in Europe being full of redheads is unclear. Based on the way selection work you could have individuals that are physically Neanderthal and behaviorally human, behaviorally Neanderthal and physically human, or anything in-between.
What exactly constitutes distinctly human behavior and how that would affect Neanderthal civilization is unclear. Neanderthals only seem to have developed art after encountering humans but then again humans didn't start making preservable art until ~40,000 years ago, and it's thought that Neanderthals lived in a harsh environment where innovation and art were luxuries they potentially couldn't afford. We have no living Neanderthals to interview to get an idea of their worldview. Ideas such as "Neanderthals were more aggressive than humans", "Neanderthals were more autistic and peaceful than humans", "Neanderthals were xenophobic", "Neanderthals had no complex language", etc. seem to be based on the sociocultural idea of Neanderthals as being the reflective "other" for humanity and don't seem to be based on actual, direct evidence (and as you can see, many of the behavioral hypotheses are directly opposed).
Yes it is entirely possible that should Neanderthals have survived, they would form an advanced society - to the same extent as our current technological society
Homo Sapiens appears to be the only animal we know of to have achieved technological means. Note however, that there is nothing particular special about Homo Sapiens: we have very little difference to Neanderthals (in fact, part of our genome is Neanderthal).
So let's say the majority of Neanderthals did not die (or morph/breed into Homo Sapiens) and instead spread throughout all continents as Sapiens did, then it is entirely plausible that their cultural and technological pathway would follow closely to that which modern humans did.
This then leads to a broader question of: What then is required to create a 'centralised government' and 'economy'? (I presume you mean equivalents of modern-day governments and economies).
After all - dinosaurs were on this planet for over 250 million years, and we do not see any evidence that they achieved an advanced technological economy.
Perhaps both us and Neanderthals are terrible at adapting to local climatic and environmental conditions, so we need to adapt those environments to us instead. We are indeed very fussy eaters, terrible at running, cannot tolerate high differences in temperature, do not give birth to hundreds or thousands of babies at a time, in fact our babies are completely defenceless and need constant attention pretty much till they become young adults (in contrast to most other animals).
As far as we can tell, Neanderthals share these deficiencies with Sapiens - and perhaps if they became widespread then, given time, they would also attempt to adapt environments to them, create farming, enabling large cities, and thus enabling art and culture, which eventually leads to complex governmental structures and advanced technology and economies.
user2352714 has provided a very good answer, but there are some suggestions based on admittedly incomplete evidence that Neanderthals and other related species like the Desisovians would not have produced a civilization like ours even has they survived in large numbers after the ice age.
The existing evidence would seem to suggest that Neanderthals were "wired" differently than the Ancestors, especially in terms of social interactions and group sizes. While the Ancestors were able to live, move and work in relatively large groups (the limit of 150 people that most modern humans can reliably closely interact with seems to point to the size of these ancestral groupings), Neanderthals seem to have only been capable of forming much smaller groups. In the intense environment of Ice Age Europe and Asia, this might have been a sensible response to the limited resources of that environment, but it also made hard limits on things like the number of skills any single band would have access to (maybe only one stone tool maker or shaman/healer per generation), while a human tribe could easily have several at any time.
It is difficult to predict what sort of society would evolve from such small groups with apparently limited potential to interact. Certainly the idea of grouping together in larger bands, tribe and eventually nations would be extremely alien, and so the ability to build large structures like irrigation canals, walls and cities would also be difficult to imagine. This essentially would be a very "low trust" society compared to something like Western Europe.
Perhaps luckily for the Neanderthals, Denisovians and possibly other related species, the Human Ancestors were the original "party animals", so we have inherited some of their traits and taken them with us through history. That is how they have made modern civilization.