It is important when answering this question to note that saying a culture is unique is not saying that it is superior. And many aspects of Western civilization often touted as unique are actually not. Another problem is determining exactly what Western civilization is, especially given the importance of Islamic culture and learning in its development.
With these caveats, it is safe to observe that several things are, in fact, unique about Western civilization. One is the importance of Christianity, which had, in one form another, enjoyed almost complete hegemony over Europe's peoples. It is the evangelical nature of Christianity that has made it unlike any religion in the world except for Islam, and as a result it has spread to most corners of the world.
Another unique element of Western culture, and one often in tension with Christianity, has been the importance of a rational, scientific worldview. While technological developments proceeded at a rapid pace in other global cultures, especially China, science as a method, a process, and a worldview has its origins in western culture.
Capitalism in its modern form originated in Western Europe. With its emphasis on individual risk and reward, it was a contributing factor to exploration, imperialism, and, of course, to the rapid development of industry. It has also helped contribute to massive imbalances of wealth, first within Western society itself, then around the world.
Finally, the notion of individual, natural rights that are inseperable from the human condition are largely unique to Western culture. These, too, were tied to economic developments (the idea that people had a natural right to property, for example, is closely tied to the development of capitalism) and they also contributed to Western notions about limited government as they developed in the eighteenth and nineteenth centuries.