What is the difference between the Son being begotten of the Father and the Spirit's procession from Him?
How can we know what the difference between them is if we don't even know what exactly each of them are in the first place?
The way I figure it, God likened the causation of the Son and the Spirit to the begetting and spiration of this world, not in an absolute manner, but as a symbol that reveals something of the reality. Given that "Begetting" and "Proceeding" are thus mere inklings of who the Son and the Spirit are, there really isn't room to say that we can know the exact difference between the two. All we can say is that there is something about the causation of the Son that is more like human begetting and something about the causation of the Spirit that is more like Spiration.
Nonetheless, it suffices to say that this amounts to there not being any real answer to your question.