Question
How do the efforts of the Jesuits in Ming China China differ from their efforts in Japan?
I know that when the Jesuits arrived in both countries, they started to convert people into Christianity and they also increased the trade relations in both countries but I do not know the difference
As user5751924 said, many differences arose due to differences in reception by Chinese/Japanese authorities – which also changed in the centuries, causing change within those countries.
According to Marius B. Jansen, in The Making of Modern Japan,
"Unlike in China, where the Jesuits chose to become intellectuals and scholars in order to penetrate the class of literati, in Japan they could concentrate on their religious tasks, as the principal intellectual class, that of Buddhist clergy, was in any case closed to them."
Here, Jansen is referring to the 16th century, but it also applies for the next couple of centuries. To elaborate...
On China, from Immanuel C.Y. Hsü's The Rise of Modern China. About Italian Jesuit priests Michele Ruggieri and Matteo Ricci who settled in Chaoching, Kwangtung, in 1583:
"Their first aim was not to win converts but to earn for Christianity an accepted place in Chinese society ... Through their knowledge of the Chinese language and culture, and their astronomical, mathematical, geographica, and other scientific achievements, they made friends with the more open-minded Chinese scholars and officials ... Ruggieri and Ricci were careful to present themselves initially as scholars and scientists; conversion naturally was the ultimate end, but they realized it could not be hurried until the ground had been prepared... Ricci and his associates were well treated in the Residence for the Tributary Envoys ... that they were allowed to reside in Peking implied imperial sanction of Christian activities in China ... Ricci's strategy was pacific penetration, cultural adaption, and the avoidance of needless conflicts with Chinese prejudices and suspicions .. it was not improper, therefore, that he compose the motets for the eunuchs, since to do so was to help the cause of Christianity. With this approach, Ricci quickly won friends and admirers amony prominent officials and scholars in Peking, among whom were dignitaries no less than a grand secretary, a president of the Civil Office, and a president of the Board of Rites ..."
On Japan, from Jansen.
"Members of the Society of Jesus, many of them members of the feudal order in Italy and the Iberian peninsula, chose to work among the Japanese feudality, confident that if they won the lords and their vassals the commoners would follow. To preach among ordinary Japanese, they reasoned, would stir suspicions of subversion. In this they proved correct; a number of powerful daimyo became converts... The affectation of Western costume, in turn, became something of a fad at many daimyo headquarters. Many of the conversions may have been expressions of this, but the constancy of other daimyo converts leaves no doubt of missionary success. Mention has been made of Konishi Yukinaga [a Kyushu daimyo], one of Hideyoshi’s leading generals in the Korean campaign; unable to bring himself to commit suicide after the loss of the battle of Sekigahara, he chose capture, mockery and execution. [he refused his followers’ entreaty to commit suicide and chose the humiliation of surrender and public execution rather than violate his Chris- tian scruples against self-destruction.]
I am aware that the Japan quotes are less relevant to the question, but from the few books I could find that mentioned Jesuits at all, this is the best I could find so far. However, if they are insufficient or unsatisfactory, I am happy to find more sources.
