The use of these titles vary based on the type of church, denomination affiliation or even the demographic makeup of the congregation. In my experience as a church music leader, I have noticed that the title of "minister of music" is traditionally used among predominately black congregations. However, "Director of Music" is used interchangeably among various congregations.
In recent years modern congregations have leaned towards "pastor of worship" or "worship leader" to describe the individual who is charge of the musical aspects of the service. Whereas, many modern churches have shifted away from using choirs during service. Instead, praise teams and/or praise bands are solely responsible for providing the music. In this case, "worship leader" may be more appropriate.
Large and affluent churches may have a Minister of Music and a Director of Music. Generally, the minister of music serves as the primary leader of the music ministry, while the director of music lead other aspects of the ministry--such as instrumental music or specific choral units.
Ultimately, there are many leaders who prefer to use "Director" instead of "Minister" especially if they are not ordained. It is more important that the leader lives a life that is pleasing to God and operates with excellence so that God will be glorified. Therefore, whatever title you choose or receive, honor God with your best sacrifice!