Why did places like India develop things like Vedantism and Buddhism while Greece developed Platonism and Rationalism? If peoples have the same cognitive equipment, why develop different philosophies?
Actually, if you look at the entire range of philosophers in different cultures, many have covered at least parallel territory. As an example, there were many ancient Chinese philosophers (largely unknown in the West) whose work paralleled or anticipated different schools of Western thought. (Bryan W. Van Norden's Taking Back Philosophy does a good job of highlighting the parallels.)
What does vary widely is which philosophies have been embraced by different cultures and why. While it may be impossible to answer this definitively, it at least seems plausible that different cultures have gravitated to philosophies that address questions that are particularly pressing for them.
As an example, in his book Guns, Germs and Steel, ethnogeographer Jared Diamond proposes that China's large, connected land-mass, without natural barriers, made warfare particularly disastrous and unbounded, and led to a society focused around peace, stability, and cultural unity. Given that, Confucianism was an ideal Chinese philosophy. On the other hand, Europe's hide-and-seek geography encouraged the constant formation of little kingdoms, and small, contained wars, and produced a less conservative society that embraced more progress-oriented philosophies.
Heidegger blames the difference you mention on the loss of the idea of Unity from Western philosophy, a loss he blames on the Greeks after Socrates. This seems reasonable to me. Roman Christianity must later have had a lot to do embedding this profound difference.
With this philosophical element missing from the Western tradition it developed a philosophy within which it has proved impossible to construct a coherent fundamental theory. Meanwhile the Vedantists and the Perennial philosophy carried on regardless grounded on an axiom of Unity and the non-dual nature of Reality.
As far as logic goes the two traditions share the same results but the interpretation placed on them by the Vedantists is off-limits for the scholastic tradition that emerged after Plato and this ensured that the two traditions maintained their distinct identities.
Differences in language, choice of metaphors, style of argumentation and presentation are often philosophically trivial and derived from culture and history. But in the case you mention it's not quite as simple as just a difference between Greek and Indian culture. There is a fundamental philosophical disagreement on how to interpret the result of logic.
For Indian philosophy the Unity of All would be the correct interpretation of metaphysics. For Greek philosophy there isn't one. Thus the only people who claim an understanding of metaphysics endorse the Indian Upanishads and everyone else has to scratch their heads at its incomprehensibility.
At the risk of being a little rude I'll suggest that the distinct difference in this case is mostly to do with how well the respective camps do philosophy.