You can trace the roots of these differences right back to the early years of the Protestant Reformation, in the 1500’s. Up until then the Catholic Church prescribed the order and content of worship very tightly.
Reformers like Luther believed that whatever the Bible didn’t forbid was acceptable in worship. So he personally saw little problem with some of the long established practices of the church, including much of the literary and ritual.
But others leaned toward a minimalist belief that only the things the Bible specifically allowed could be used in worship. Without discussing the merits of those two views here, one way or another, they had a big impact as follows:
The minimalist approach drew from the more spontaneous descriptions of New Testament worship in 1 Corinthians 14:26,
...When you come together, each of you has a hymn, or a word of instruction, a revelation, a tongue or an interpretation. Everything must be done so that the church may be built up.
And from Ephesians 5:18-20,
Do not get drunk on wine, which leads to debauchery. Instead, be filled with the Spirit, speaking to one another with psalms, hymns, and songs from the Spirit. Sing and make music from your heart to the Lord, always giving thanks to God the Father for everything, in the name of our Lord Jesus Christ.
The different spiritual gifts were used by a variety of people in the church. This could be done with a planned order of worship (1 Corinthians 14:40) or as the Quakers (Friends) interpreted, spontaneously as the church gathered.
This simpler form, which emphasized evangelism and preaching, became the style of the Puritans, Congregationalists, Presbyterians, Methodists, Baptists, Mennonites and others.
In America the open air preaching of the Methodists, the Great Revival of George Whitfield in the Colonial days, followed by the “Camp Meetings” in the pioneers days, then evangelists such as Charles Finney, Dwight L. Moody, and others--led to a style that incorporated spontaneous prayer, lively congregational singing, as well as a passionate appeal for repentance and personal faith in Christ as Savior. This trend continued into the twentieth century.
In the Jesus Movement of the 1970’s Christian lyrics were blended with rock style music as an outreach, bringing hundreds of thousands of students and younger adults into the church. This entire evangelical movement emphasized going where the people were, proclaiming the gospel, using familiar sounding music and relevant preaching.
Eventually this style carried over into many churches and is the basis of the contemporary style of many churches today.