Burdensome though it may seem, these rules exist to ensure that writers and performers get paid fairly for their work. Making a living as a musician or writer is a precarious existence that relies on the royalty system for income, keeping a roof over their heads and food on the table.
It is a complex area with several different components.
A work of any sort remains in copyright until 70 years after death of the original author or composer. After the 70 years have expired, the work passes into the public domain and can be used free of charge. For a song, music and lyrics may be considered separately, so if someone writes a new tune to old words, the music remains in copyright even when the words are in the public domain. It is also possible to copyright an arrangement of a work, even if that work is now in the public domain.
Copyright means that you cannot reproduce the sheet music, printed words, projected words, etc. unless you have the appropriate paid licence to do so.
The author also holds performance rights on their work, and are entitled to receive a royalty whenever the work is performed. The author’s rights are covered by the Performing Rights Society and you normally require a PRS licence to perform a work in copyright.
Separate to the author’s rights, whenever a work is performed the performer also has rights to receive a royalty. Performers rights are covered by a Public Performance Licence and you normally need a PPL to play commercial recordings.
Acts of Worship
There is an exception to the PRS and PPL which allows you to use copyright works and commercial recordings as part of an act of worship. This is true whether or not you have a CCLI licence. However it only covers services and not other social events held in church. We strongly recommend looking at the CCLI licences for this sort of use.
This exception only covers works for use within the church building itself. It does not include recording, broadcasting or streaming, even if you record a live service in the church building.
There is a good summary of what this means in practice in this post by Resound Worship.