The fundamental difference between India and China is that China is a nation-state while India is not. Keep in mind that the definition of a "nation" is NOT synonomous with "country". A "nation" is any community of people with shared characteristics. Some nations don't have states, such as the Jewish people before the creation of Israel. Nations that do have states, and dominate that state, are called "nation-states". An example would be Bangladesh, where 98% of the population are Bengalis. Similarly, China is also a nation-state with over 90% of its population being Han.
India, on the other hand, is not a nation-state. There are literally hundreds of nations that make up India, from Telugus to Punjabis to Gurkhas. India is a civilization-state, where its people, despite having regional cultural differences, are all members of the unique Indic civilization. China is also a civilization-state in addition to a nation-state, because the Sinic civilization, for the most part, is self-contained in China.
Most other nation-states are parts of much larger civilizations. Take for example France, Italy, and Spain. All have their unique cultures, but they also share important civilizational aspects with one another, such as the Roman Catholic religion and languages based on Latin. The legacy of this "Western" civilization can be ultimately traced back to the ancient Greeks and Romans.
So, in general, we see the hierarchy of identity as something like this:
Civilization/Culture -> Country/State -> Nation/Ethnicity -> Class -> Family -> Self
What China is trying to do is bring seperate nations (Uighurs, Tibetans) under the sphere of Han civilization; more specifically, under the sphere of the Han nation-state. It can do this because the vast majority of its population are Han, and the assimilation of such small minorities into the mainstream is not only possible but also desirable. As Sir Ray has pointed out many times, much of the history of China was the history of Han cultural imperialism, as Han civilization spread from its original home in the Yellow River Valley to occupy much of the land that is today China.
India, on the other hand, does not follow the policy of cultural nationalism that is prevalent in nation-states like China, but rather liberal pluralism. Each Indian "nation", if you will, is part of the much bigger whole that is India and Indian civilization. India does not seek to assimilate its people into one, but preserve diversity with unity.