How did the fundamental split of the Roman Catholic and Greek Orthodox churches influence later history?
It is not certain how much this schism really affected later history. The East and West would have grown apart even without the schism and it is not possible to know how much the schism contributed to that sense of difference. It would be a good idea for you to check what your specific textbook and/or instructor have to say on this issue because different people can give different answers.
I would argue that the major effect of this schism was to widen the gap between East and West. For example, the schism helped to cause tensions between Normans in Western Europe and the Byzantines. This had serious consequences later on. This was because the Byzantines had to ask Pope Urban II for help against the Seljuks in the 1090s. This led to a series of Crusades in which Normans and other Westerners came through the Byzantine Empire. Contact between the two disparate religions helped lead to more tensions. These came to a head in 1204 when Crusaders actually besieged and sacked Constantinople. This might not have happened had it not been for the schism.
We can also say that the schism helped to build the divide between Eastern Europe and Western Europe. This divide became very important in our own time when it was part of the Cold War. By the 20th century, Western and Eastern Europe had been growing apart for centuries. The fact that they had different religions helped to push them apart. This meant that the two saw themselves as parts of different civilizations. This helped to cause the divisions that would become even more crystallized by the Cold War and by the legacy of communism in the East.