'China's' is a genitive form, meaning 'of China.'
'Chinese' refers more to the nationality, language and the culture of China.
For example, you would say 'I saw a beautiful Chinese painting in the art gallery'. But you might see the phrase 'China's trade surplus' in an article about the world economy. This has nothing to do with nationality or culture, it is simply 'the trade surplus of China' - just a figure relating to the country.
In most cases, we would use 'Chinese', for example 'This is a beautiful Chinese silk scarf'.