Well, in my opinion, Jesus said the Father was greater than He because the Church teaches that while Jesus and the Holy Spirit are both God, the Father is the ultimate “head” of the Godhead Trinity. For example, Christ said that even He doesn’t know when the Father will renew the Earth, send Jesus back, resurrect the dead, etc. on the Last Day. Which doesn’t mean that Christ and the Spirit are lacking, because they are both God, and God as a whole is not lacking in any way. But, in answering your second question about duties, the individual persons of the Trinity are different, though the same God essence. Their difference comes in their specific attributes, which are necessary for God to be fully and perfectly God, but could not properly exist in only one person. For example, if God were not Father and Son, God would not have a relationship within Himself; He would be alone. But, in order to be even more perfect than God would be if He were alone, He consists of a Father and a Son, who while both God, form a familiar relationship, and their love is expressed as the Holy Spirit. If God were just God, as in Islam and some other monotheisms, He would not be as perfect as the Trinitarian God.
Always remember, in Trinitarianism, God is an essence, composed of three persons. In polytheism, deities are individual persons called gods, each being a god. But in Trinitarianism, being God is being a specific sort of being, which is a Divine being. The God-essence is a set of characteristics that makes something perfectly and completely divine, and there are three individual persons that are God: the Father, Son and Holy Spirit. Just as Bob, Joe and Sarah would all three be human, the Trinity are all three God. Not distinct essences or souls from one another; not individual gods. Just God, in three persons.