What are differences and similarities between eastern and western religions between 1000 BCE and 750 CE on how they influenced socio-political events?
This is a fairly broad question, but to start with we might divide the religions that developed in this period into three groups:
1. Near Eastern (Judaism, Christianity, Islam): all three monotheistic, and all three associated with particular kingdoms (Israel/Judea for Judaism, Roman Empire after the 4th century for Christianity, the Islamic Caliphate for Islam)
2. South Asian (Hinduism, Jainism, Buddhism): All three developed beliefs that people can practice meditation to transcend their lives and unite with something greater. Of the three, Hinduism had the most developed system for organizing society - the caste system. Buddhism developed the first monastic institutions, and all three religions taught their followers that in order to seek transcendence, they must leave society and wander, begging for food and sustenance. They all also taught the importance of reincarnation and that your experiences, good and bad, are the result of your deeds (karma) in past lives.
3. East Asian (Confucian, Daoist, and some forms of Buddhism). Confucianism and Daoism were both concerned with the best way to achieve harmony in society, and differing Chinese dynasties followed Confucian teachings of moral reform, and Daoist teachings of letting things run themselves. Confucius taught that the highest moral good was "ren," (benevolence or empathy), and that if one has ren one will act properly - a virtue called "li." He also taught that there is a hierarchy in society, on which harmony should be based: wife should follow husand, younger should follow elder, subject should follow emperor.
You might try checking out the link below for information on these and other religions in the world.
This is a large span of time. Between these two eras, you have the creation and destruction of religions and societies. A difference you could focus on between Eastern and Western religious influences on socio-political events would be the styles of religious leadership and how they translate into the social activities of the people they led. The Mayans sacrificed people with religious leadership and the people in the East followed Confucianism, despite Confucius not being a god (most religions focus around a deity or multiple deities). A similarity that could be focused on between the two would be the creation of government control based on religion, such as nations that have been led by the Catholic, Jewish, or Muslim faiths.
I've included a couple of links for possible research for you. Remember to cite your sources!