Theologically, there is basically agreement. Everyone can agree on "proceeds from the Father THROUGH the son", even if no one puts those words in the Creed because both Churches understand that the original source of the Spirit procession is the Father, and yet the Son as a mediate channel of this procession. The Son plays a role, but the role is different. For example, the biblical reference is John 15:26 "When the Advocate comes whom I will send you from the Father, the Spirit of truth that proceeds from the Father, he will testify to me."
Now we come to the wording. The original Nicene Creed had neither, but the words under debate (not the Filoque but ANY words referring to the procession of the Holy Spirit) were added at the Council of Constantinople. The words were added to combat the heresy of Macedonianism, which believed that the Spirit was a creation of the Son, subservient to Father and Son. It was used at the Council of Chalcedon (together with the original Nicene Creed.)
There is (apparently) a subtle difference in the word meaning in Latin and Greek, both of which translate to "proceed". In Greek, the word can only refer to coming from its ultimate source, where in Latin it can come from a mediating channel.
The Latin Church adopted the Filioque wording in order to stress the equality and coopertion of the three Persons (in opposition to Arianism, for example, which rejected that the Son was equal in Divinity to the Father). The Greek Church did not use the Filioque because they wanted to stress the distinction of the three Persons (in opposition to Monarchianism, that says they are just three characterizations of the same God).