Hey Lydia!This is the question that many people have pondered in their minds. I hope my explanation will help you to sort things out!
A Minister of Music is one who has the gift of being able to musically pull things together write charts, lead sheets, play and instrument etc as well as run the music department within the church. This is not the case in some churches - sometimes the Worship Leader will be in charge of the department or the person in charge of the music department will sometimes be called the Minister of Music because many churches do not sing Praise and Worship choruses or songs that would lend themselves to this contemporary style. It is strange but some just understand what you do when you say Minister of Music instead of Worship Leader.
Many churches have a song leader or worship leader who heads the department and the worship leader will be charged with the responsiblility to lead the congreation in worship through song...with an emphasis on Praise and Worship style. This person has no musically training as far as a instrument or formal musicial training. In that case there will be a Chief Musician to help assist the leader.
I hope this helps...If you would like more information on the job descriptions try getting your hands on "Exploring Worship" by Bob Sorge or "How to have an Effective Worship Ministry" by Tom Kreuter that will give you a start.
If it would help you anymore - I have written an article called "Hidden Opportunities" on this website. Perhaps it will shed some more light.
Blessings on you...Debbi