The difference between the two expressions is insubstantial, for they convey the same meaning, as in two English expressions 1) They claim about him that he is a German spy and 2) They claim him to be a German spy.
The idea is that John is older than Jesus by 6 months, so he came earlier than Jesus to the world, however Jesus pre-existed as eternal Logos of the Father his physical/biological birth from the virgin Mary. In the very words of John only pre-existence of Jesus is asserted, but in the light and the entire context of John's text and theology, John the Baptist must have indicated also the eternity of Jesus.
This can be seen also in Mark 1:7, when John the Baptist, whom Jesus acclaims to be the greatest of all prophets and even all humans who ever lived (Matt. 11:11), says that he is unworthy to stoop and untie his shoe-laces, and saying that he, John, is just a preparing people through the baptism of repentance before coming of the baptism of salvation through the Holy Spirit and God's consuming fire (Matt. 3:11), that is to say, God's grace that works salvation in human hearts (Col. 1:29); now, as the fountainhead and the principle of the preparatory baptism of repentance is John the Baptist, the highest amongst the men whoever lived in the earth, so the Jesus is the fountainhead and dispenser of the baptism of Salvation, of the divine uncreated fire, that is to say, divine working and operation in us, indeed His own life and working in us (Gal. 2:20), that consumes the ὁλοκαύτωμα of our sinfulness and recreates humanity into the new creation (2 Cor. 5:17). That is why John is dreaded when he sees Jesus coming to be baptized by him, for he knows that it is he, John, who needs to be baptized by Jesus (Matt. 3:14).
Thus, "He is next to me, and yet prior to me" should be understood in this deeply theologically clad meaning.