The Pulpit Commentary* says that the word γενόμενον (KJV: made) in the phrase γενόμενον ἐκ γυναικός (made of a woman) in Galatians 4:4 implies a previous state of existence, whereas "born" (so NASB, HCSB, NRSV, ESV, NIV, etc.) does not.
. Made of a woman (γενόμενον ἐκ γυναικός); made to be of a woman. This, indeed, was probably the sense intended by King James's translators, when they followed Wicklife and the Geneva Bible in rendering "made of a woman;" whilst Tyndale and Cranmer, followed by the Revisers of 1881, give "born of a woman." Just the same divergency of renderings appears in the same English translations in Romans 1:3, "made of the seed of David (γενομένον ἐκ σπέρματος Δαβίδ)," except that Tyndale has "begotten" instead of "born." The difference in sense is appreciable and important: "made" implies a previous state of existence, which "born" does not. So far as the present writer can find, wherever in the New Testament the Authorized Version has "born," we have in the Greek either τεχθῆναι or γεννηθῆναι: γενέσθαι never having this sense at all. As in Galatians 3:13 (γενόμενος ὑπὲρ ἡμῶν κατάρα), "Being made a curse for us," and in John 1:14 (ὁ Λόγος σάρξ ἐγένετο), "The Word was made flesh;" so here God's Son is described as "made to be of a woman," the phrase, "of a woman," being nearly identical in import with the word "flesh" in St. John, distinctly implying the fact of the Incarnation. The preposition "of" (ἐκ) denotes derivation of being, as when it is found after the verb "to be" in John 8:47, "He that is of God;" "Ye are not of God," pointing back to the claim which (ver. 41) the Jews had made that they had God for their Father. The construction of γίγνομαι, to come to be, with a preposition occurs frequently, as in Luke 22:44; Acts 22:17; Romans 16:7; 2 Thessalonians 2:7. There can be no doubt that γενόμενον must be taken in the next clause with the same meaning as here.
Does the word γενόμενος indeed indeed imply previous state of existence? Is this better represented by the English "made" rather than "born"?
*Huxtable, E., "Galatians -- Exposition," The Pulpit Commentary, vol. 20 (Eerdmans), 1962.