To be brief, mentioning one Person of God as responsible for creation of Heaven and Earth, another Person as by whom all things were made, and another as the Person giving life is not meant to exclude the other Persons in these activities. In fact, the whole Trinity participates in these three activities; nothing in the Creed excludes the other Persons from them. I may be incorrect on this next count, but I think that these three activities are connected to their respective Persons in the Creed because they are, in a sense, iconic (or representative) of who the three Persons of God are and what their relationship to one another means. The actual meaning of their ontological relationship is revelation and cannot be explained in words, but I think that tying the Father to the foundation of the being of all creation, the Son to the fabric or Logoi of creation, and the Spirit to the life or motion of creation (most importantly, of actual living things) is specifically given in the Creed as a kind of glass-darkly analogy to who these Persons are and how these persons relate to one another.
In Orthodox theology, the distinguishing is spelled out in the Creed also, namely that the Son is begotten of the Father and that the Holy Spirit proceeds from the Father (going from the Greek wording always commonly held in the East and West; the Latin adds "Proceeds from the Father and the Son" in a sense slightly different than the Greek). The Father is not given such a relationship explicitly in His clause of the creed, but it can be immediately deduced from the other two clauses: the Father is that Person from Whom the Son is begotten and from Whom the Spirit proceeds, i.e. the Father is the Person of God Who is in a sense the ground and font of the eternal, uncreated presence of the other two Persons.