Do remember that Taiwan is also just part of China. It used to be the imperial army that after loosing to the communists, needed a place to hide out. Taiwan was a good space as it has lots of natural defences.
From 1895 till WW2 it belonged to Japan (Before this it belonged to China a bit, but from the 15th century till about the 19th it was pretty well..... DUTCH (Yet most trading with China took place at Dejima Island in Japan). After it and its people were handed over to Japanese rule by those whose ancestors now still reside on it. The Chinese took it back after WW2. The ROC which then ruled China after loosing the civil war to the communists, fled with 2 million soldiers to Taiwan (basically killing anyone from Taiwan that opposed them). With that they took most of China's treasures and gold reserves (many of these are still locked away in bunkers within the mountains of Taiwan). About 140 000 local Taiwanese people were killed during the Republic of China's invasion of Taiwan.
You have to consider modern day China.
Hong Kong used to funnel 90% of the money coming into China, today its barely over 10%. The government has also worked dramatically on poverty reduction, with extreme poverty eliminated. Generally in part China is on the UP still. Sure its very different, but what they have achieved is amazing.
Only the following countries recognize Taiwan
They cant become a fully independent country as the UN Security council needs to approve them for this. Currently Russia and China has said firmly NO.