Even today, the Pope does not always have the last word. Quote frequently, the Pope issues the doctrinal statement, and someone else issues the disciplinary statement.
Acts 15 simply shows a prototype of the way the Church does things today. For example, recently, Pope John Paul II issued an encyclical called “Ecclesia de Eucharistia” on the Holy Eucharist in the life of the Church, while touching only very slightly on liturgical abuse. For the most part, it was a doctrinal, reflective, and highly theological document.
But after that, and to some degree in response to it, Cardinal Arinze (or the CDW) issued the instruction “Redemptionis Sacramentum” detailing the abuses that were to be stopped and how to go about it.
In much the same way, Peter was giving the “encyclical”, while James gave the “Instruction”. The text of Acts 15 shows the difference in content between the two pronouncements. Peter gives the doctrine, James affirms it (much like Cardinals and Congregations do today) and gives a disciplinary pronouncement.
As for the “falling silent” part, it is not to be taken alone, but contrasted with the earlier reference that there was “much argument.” The silence came only AFTER Peter’s pronouncement, which Paul and Barnabas backed up.
