Someone better versed in ancient history may be able to offer an authoritative answer, but I can at least offer a biblically-grounded one.
It may have had to do with the entry of Gentile believers into the church. Acts 10:1-11:18 relates the first significant disagreement resulting from this development (Peter's acceptance of Cornelius and his household), and Acts 15:1-35 relates the debate and council decision as to what the Gentiles "needed" to do, to be accepted into the church as saved brothers. A significant party within the Jewish brotherhood asserted that the Gentiles had to submit to the entire Law to be saved; Peter, Paul, Barnabas, and James however argued that it was faith in Christ that justified all men in God's sight. Quoting in part the letter that announced the council's final decision:
28 It seemed good to the Holy Spirit and to us not to burden you with
anything beyond the following requirements: 29 You are to abstain from
food sacrificed to idols, from blood, from the meat of strangled
animals and from sexual immorality. You will do well to avoid these
things.
In a later writing to the church at Corinth, Paul gave an instruction that only people who had been observant Jews at the time they became Christians, needed to or should continue to observe the entire Law. Gentile believers needed not do so, and in fact should not do so:
17 Nevertheless, each person should live as a believer in whatever
situation the Lord has assigned to them, just as God has called them.
This is the rule I lay down in all the churches. 18 Was a man already
circumcised when he was called? He should not become uncircumcised.
Was a man uncircumcised when he was called? He should not be
circumcised. -- 1 Corinthians 7:17-18
Hence, the majority of modern Christians (being non-Jews) would not and indeed should not observe the wearing of tefillin, nor any of the other ceremonial aspects of the Law.
