Jaroslav Pelikan gives a nice turn of phrase about the transition from Jewish Christianity to Gentile Christianity:
The leaders of [the early church] were Christians of Jewish origin; despite their differing answers [re: the Acts 15 issues], they asked the question of continuity between Judaism and Christianity with a deep personal poignancy. As converts began coming more from pagan than from Jewish ranks, the poignancy lessened and the obverse side of the question became more prominent. For Jewish Christians, the question of continuity was the question of their relation to their mother; for Gentile Christians, it was the question of their relation to their mother-in-law. What was offensive about Christianity in the eyes of Gentiles was, to a considerable extent, what it had inherited from Judaism. [He then goes on to cite criticisms from Celsus, Marcion, et al.]
The Emergence fo the Catholic Tradition (100-600), Univ. of Chicago: Chicago, 1971, p. 14.