What’s the difference between the Father’s relation to the Son, one of “begetting,” and His relation to the Spirit, that of “proceeding” (John 15:26). A distinction without a difference, serving only to protect against the conclusion that the Spirit is another Son?
Augustine’s discussion of this distinction comes on the heels of his introduction of the idea of Spirit as “gift” (The Trinity, 5.3.12), and it depends on this identification. The Father produces the Son and therefore is “the origin” since the produced is origin “with reference to what it produces.” But is the Father the origin of the Spirit? To says so would be to say that the Father is “origin not only for what he begets or makes, but also for what he gives” (5.3.15). He denies that the Spirit is another son because the Spirit comes for “not as being born but as being given.”
The fact that the Spirit is gift also explains how He can be both the Father’s Spirit and ours: “what has been given is referred both to him who gave and to those it was given to; and so the Holy Spirit is not only called the Spirit of the Father and Son who gave him, but also our Spirit who received him” (5.3.15).