The first thing to realise is that what's termed an "autonomous" region doesn't necessarily mean what you think. You might look at that map and think "Oh, the Tibetans and Uighers just rule themselves" but that's not really true. Yes, they can make their own laws, and central laws don't necessarily apply, but both areas are still run by Communist Party officials loyal to Beijing (a lot of whom are Han, though there are locals as well... assuming they're willing to sign a document repudiating their traditional religion). Hong Kong has far more autonomy than Xinjiang. In fact, in some ways Tibetans in Yunnan province have more freedom than in Tibet.

For a very crude parallel, think of them as something like States in America. They can make their own laws on some things, and not others. They're a lot less autonomous than Puerto Rico.

Now, saying this, there are a couple of other things to remember. The United States is pretty much, culturally-speaking, the same everywhere. Every state has, traditionally, been controlled by White, English-speaking Protestants. But 60 years ago, Xinjiang had very few Han; it was mostly Muslims from various Central Asian and Turkic ethnic groups. China's far-flung regions are very different from the eastern regions. Different ethnicities, languages, religions, and cultures. It's also important to remember that 120 years ago, China wasn't even controlled by Han, but by Manchus, who deliberately stopped Han from moving to certain regions in order to keep the local cultures intact (and thus loyal). The Tibetans, Mongols and Uighers were all part of the Qing Empire, but they weren't really under the control of the Han Chinese in the way that, 120 years ago, Georgia was under the control of White, English-speaking Protestants.

China is huge, and has always been difficult to keep together. The fact that the Chinese government is able to control a region of Central Asian Muslims 2000km who were de-facto independent during the Warlord region (and actually independent in the Ming dynasty) isn't a sign of Beijing's weakness, but its power. Of course, they know it's difficult to control such a non-Han area, which is part of why the government has been pushing its "Go-West" policy, encouraging Han to emigrate to the Western regions. Nowadays, the population of Xinjiang is about 40% Han. (This is not the only reason: there are plenty of non-political reasons to encourage people from the developed but extremely over-populated East to move to the West.)

As to its claim over Taiwan, that's just a relic of the Civil War, which it's important to remember is not that long ago. The KMT ruled China. The CCP rose up and drove it out from the mainland. But attacking Taiwan was too difficult, so it's been left in limbo. It's probably worth noting that the Chinese Communist Party claims to be the legitimate government of all China (including Taiwan), and the KMT (which is still one of the two main parties in Taiwan, and was until recently the ruling government) also claims to be the legitimate government of all of China. Both wish to unify the mainland and Taiwan, as they view them as China, it's just that they have very different ideas about what that China should be like. In fact, the previous President of Taiwan, President Ma, insisted that he was the President of China and his capital was Nanjing, with Taipei just a "provisional capital". (The other party in Taiwan, which recently won the election, does not view Taiwan as part of China.)

To conclude, the reason China and the US are very different is because the US is actually much older, as a state. The current Chinese republic was founded in 1949, only 67 years ago. 67 years after its founding, the US had pretty poor control of its western regions.