You could really think of a "Music Director" the same way you would think of a "Music Producer" the difference being the Music Director is playing your music along with you on stage. While the primary role of a Music Producer is to make sure your songs are recorded the best way possible in the studio, the Music Director is the person who makes sure the songs you recorded are reproduced faithfully in a live setting.
Even though many artists may possess lots of musical talent, it can be very difficult for them to convey thoughts and feelings about how their songs should be played live. Couple that together with the highly technical nature of actually playing musical instruments and speaking the language of music to musicians, this could become a whirlwind of despair for an artist who isn't completely comfortable in this element. The Music Director takes the burden off the shoulders of the artist/performer and allows them to focus more on the performance without having to stress about all the details that arise with it.
There are many different minute details that typically need to be thought through, organized and planned to execute the intricate performances that are typically expected of artists today. Music Directors on today's stages are "wearing multiple hats" and must have mastery of many different disciplines. From writing and arranging (or sometimes re-arranging) pieces of music to fit a certain time slot to operating digital audio workstations live on stage for playback, the Music Director is the person who is making sure everything is going as planned, the band sounds great, the playback sounds great and most of all the artist is comfortable enough to deliver the best performance possible.
The Music Director may often be tasked with finding other musicians and background singers for your band, administrating travel arrangements, dealing with an artist's label or management, talking with event and venue staff, or even coordinating with the audio engineers and sound and rental companies. Even if the artist is an accomplished musician themselves, having a Music Director you trust to get the job done night after night and gig after gig could be viewed as a necessity for any serious artist.