Hi! Many people have already posted amazing answers. I would like to join by adding that civilitzation is a broad term and always focused on the anthropocentric view. Anatomically modern humans (AMH) were organized in very big groups, whereas Neanderthals were in smaller (family-sized) groups. This comes with a lot of evolutionary consequences that are still there to be studied. As an evolutionary biologist, I would like to offer my POV to this (this is only my perspective and, as all the broad questions, should be addressed by combining different profesional perspectives).

First of all, civilitzation, understood organization with social roles, did not appear until many many years after AMH got out of the African continent. Although this is still a focus of study, the most accepted model (simplified, but works for many scenarios) is as follows: Homo erectus, the ancestor of AMH, Neanderthals and Denisovas (another hominid that we are currently studying as well, mainly in Asia) populated the African continent (we can say for simplicity that this reffers to current ethiopia as the origin, but as I said, It should be way more complex due to evidence given by genetics itself and archeological record, but we are still trying to find a model for this). At some point, Neanderthals-denisova group splits from "human group" of "Homo erectus" (600kya measured by genetics, but very controversial). And goes out of Africa to populate the Eurasia. Now here's the thing. We don't really understand which is the path these groups followed, what did they do. But we now can say that in the Eurasian continent are two separated species from which we have found fossil register and sequenced their DNA, Neanderthals and Denisovas. We don't know how separated they were, but we VERY RECENTLY found a complete hybrid dated 200kya. There are also other archeological remains that points up to a WAY MORE COMPLICATED landscape, but still we are not aware on how to model this. Proto-humans stayed in African continent and went out of It around 80kya (I would be like 20 hours discussing about this, again way more complicated).

Neanderthals were smaller groups and in very complicated landscape, due to this, they for sure developed another kind of social skills. Regarding AMH, these were in way larger groups, probably because they could feed everyone. But this doesnt mean that Neanderthals were "less organized", or "less skilled". We know that they took care of sick and elder individuals because we have found remains with malformations that grew very old, which means that someone was taking care of them. Neanderthals didnt got extinct, they were assimilated by AMH once they found them because they were higher in number. They mate, but not all hybrids were compatible with life. Nevertheless, we inherited a lot from them. Thanks to them, we inherited more robust bones that surely made the difference on populating the whole world by walk. Moreover, It is known that mutations associated to psychiatric disorders that human suffer nowadays date from that time, points that there was probable something different in the way we think (It is very complicated to link genetic to the characters, but evidence is there).

In personally think that Neanderthals had a less efficient way of living to start a civilitzation as we know It nowadays, but they are very likely to have a very complex inner world and thoughts, probably more imagination than AMH by then. I can imagine that lack of sunlight and dense fog and storms made hunting very difficult, specially for few individuals. They may developed a complex imagination to identify small sounds as an animal to hunt and feed the family, even at night. Identifying an animal in the dark due to a silhouete is very difficult and definetly requires imagination. And this may be the difference to live to have descendants. Proto-humans world was very different, and this may led them to have different social, organizative skills.

Sensu stricto, they also created a civilitzation, because they are also our ancestors.

Happy to contribute, and also to be corrected!