My thoughts on this developed from several different areas which eventually intersected, from conversations with my father to reading Closing of the American Mind to the study of Greek, Roman, and early Christian and ancient Jewish cutlure. There are a few things I have in mind when I speak of consequences beyond simply an inability to understand references within art, literature, etc. Perhaps the most important is morality. In my experience (personal interactions, reading the thoughts of others, reading history and the historical texts) and in what I have read in the sociological literature, the conception of "culture" in the "West" as well as the influence of Christian thought/morality has created for most a false sense of the basis for human morality, which is compounded by the commonality among "different" Western cultures compared with regions which have non-christian traditions dating back for centuries or millenia. The result is a mindset that "we all pretty much think the same" when in fact this is mainly an illusion due largely to Chrisitian influence on the West. The differences between Italian culture and French culture and American culture can be great, certainly, but one need only look at the initial failure (and eventual "solution") of pharmaceutical companies marketing anti-depressants in China, or the problematic extension of DSM criteria to South American countries, or even the issues Special Forces had training Saudi troops to see how radically different worldviews can be in places without centuries of Christian influence. This is only more true now that most of the countries which were so greatly influenced have such greatly decreased Christian populations.
As for how this can have truly wide-ranging consequences, look at the mission to "spread democracy" and other European/American ideas. Or take science: Bertrand Russell wondered, with his Eurocentric perspective, wondered why China did not develop science (meaning the systematic endevour to investigate and understand the world, and the development of methods and instruments to do so) because they did not have what he regarded as the main impediment to such a development: a religion like Christianity. What he failed to realize (because of his Eurocentric perspective) was that very few cultures have had anything like the impetus necessary for the development for science, or the fact that this requires a particular worldview with certain elements: a belief that the world/cosmos can be understood because it has some sort of order, that there is a reason/good in such an endeavour, etc. Some cultures have come close, and may have continued had they not been conquered or for other reasons, but these are still few and far between. Christianity helped create (after initially stifling, and then later impeding) the development of science because after the fusing of Greek philosophy and Christianity the Christian intellectuals viewed God as working with a plan, with a certain order, and to some extent knowable (compared to, for example, the way many groups falling under the modern categorization of gnostic saw god). The scholastics extended this into an attempt to understand god and god's works through reason, logic, and higher learning, and the early modern philosophers, scientists, and mathematicians further extended it throught the investigation of the natural world.
Chinese philosophy did not provide the pre-requisites to encourage that kind of investigation and development. Not understanding how worldviews can radically differ with so many ramifications makes possible (or likely) truly poor international relations.