If Asian students get good test scores in math and science while studying in their own countries, why do they come to the West to study and why don't Westerners study in Asia?

They leave their countries for better opportunities and facilities which are not available in their home land.

The reason why many Asian students come to Western countries is exactly because of that-- test scores. Asian education systems revolve around high-stakes, standardized tests, and emphasize classes that have relatively comparable curricula (math and science). Tests in math and science can be easily compared between different regions of a country, or continent for that matter. Many Asian universities are lecture-test based, with little teacher-student interaction. As a result, students spend hours laboriously pouring through textbooks and practice tests. The pedagogical methods of Western academia focuses heavily on reasoning, thought, analysis, and discussion, particularly prevalent among the Liberal Arts. Many International students at my school from Asia are surprised to hear things such as 'quiz section', 'writing section', or 'group discussion' exist.  They're not used to being able to participate in class, being able to reason with a professor, being asked to write an essay or give a presentation in a STEM-based course. One of my friends from Japan told me how students at her school (one of the top in Japan) rarely attend lecture, and opt for internships or work instead. Prior to semester exams, they attend cram schools and pour over textbooks for one of the few things they'll be graded on-- tests. Even though classes in STEM fields in the West might still focus on tests and standardized curricula, the methods in which it is conveyed-- through discussions, and understanding why certain concepts are being taught to provide the foundations for higher-leveled courses-- is something largely omitted from elsewhere. This certainly doesn't take into account the drastic change in atmosphere between teacher-student relations of the East and West, or the college lifestyle, but together, Western education presents a refreshing alternative to Asia.

There are two kinds of know-how, namely Theory and Experience. Theory is well studied and interpreted. You don't need to come to Western countries to learn them. For example, you don't need to go to Cambridge to learn Newton's theory. Experience is not well studied. People know it works, but don't know why. If you stay in Asia, you don't even get the chance to see the problem, not mention the experience of addressing it. High school / college math are simply a collection of well studied theories. You can study them everywhere. However, if you want to become a professional mathematician, you need to learn it from academic advisers who are in fact senior mentors with rich experience. You have to find top 0.1% adviser in China to learn what you can learn from an average adviser in US.