People seem to get confused about the difference between a voice coach or a voice teacher. Justifiably so. Often a voice teacher does a tremendous amount of coaching when rehearsing a song with a student, and a voice coach will often instruct a student on a vocal technique issue that will help deliver the song more effectively. I'm sure many people have lots of passionate opinions about the differences between a coach and a teacher, but these are a few of my thoughts on the subject (which I think are pretty good!)
A good voice teacher will instruct a student on fundamental singing techniques. How to produce beautiful, interesting, energetic vocal sounds that will be able to master a variety of vocal styles and effectively deliver songs with ease and a full palate of emotional expression. The best teachers are familiar with a variety of vocal approaches and have an understanding of which approaches will best unfold a singer's voice for the songs being studied. This must be done with great care so the singer will develop a fuller knowledge of how the voice works and how he/she can help it be "allowed" to sing really well, regardless of the musical genre. The very, very best teachers are also somewhat intuitive about the singer's inner being - what are this person's vulnerabilities, passions, strengths, weaknesses, and the true goals of this person - to know the best ways to help the singer work with who they are to achieve what they want.
A good vocal coach puts the polish on the singer. A coach will work a song with a singer until it becomes flawless (ideally...). The singer must understand the intention of the composer, how the piece relates to the whole work (if it's from an opera, musical, oratorio, etc.); the timing, rhythms, pitches; the language and dialect; emotional expression; how much liberty can be taken with a song - how much SHOULD be taken. A good coach covers all of this, while also acting as another ear to ensure the beauty and resonance of the voice itself is not compromised. Additionally, a good coach will help chose repertoire for a singer. A versatile coach will be able to cover a variety of genres, but there are incredible coaches that just specialize in one field, like many operatic coaches. These are for the professionals. You'll know when this is the one you want...
As I said earlier, there's a lot of overlapping within these two specialties, especially with teachers of beginning and intermediate students. Essentially, though, what are the greatest strengths and the most intense focuses of the trainers. Regardless of what one calls himself, he's got to know a lot about singing and music to be responsible for training a vocalist!
I'd love to hear other opinions about this! Shout out and add to this list, if you'd like!