While searching for universities to do my Ph.D. in mathematics, I noticed that most top universities in India use incredibly competitive entrance exams (much more difficult and time constrained than the GRE/TOEFL/Subject GRE) and only after you've cleared those and based on your ranking in those do they ask you to provide them the transcripts, recommendation letters, Statement of Purpose etc. indicating that the results of those (mostly objective) entrance exams are considered more important than the other three components of your application, mentioned above. The exam is usually followed by an in person interview which is equally rigorous. Even if they don't conduct the exam themselves, they rely on some other national level entrance exam as a screening tool and/or as a way to create a merit list to offer admissions.
However, in the US and most of Europe (I suppose) universities only require you to have done well in some (comparatively) easy exams like the GRE/TOEFL/Subject GRE and many even waive that requirement off. A university there using their own entrance exams and interviews as a screening process is unheard of and the admission decisions are made based on transcripts, recommendation letters, Statement of Purpose etc. The qualifying exams before one is allowed to officially begin their Ph.D. allow a couple of years of preparation thorough coursework in the university and though tough, I think they allow some flexibility in terms of when the student wants to take the exam and what electives he's opted for. Even if one fails to clear these exams some universities allow the student to terminate the program early after the coursework with just a master's degree. (I'm not 100% sure about the last few sentences.)
My question is that what are the possible reasons for such a drastic difference in the admission process?