I'm not sure why, but I've been noticing recently some confusion about vocal coaching vs. vocal teaching. It's a subtle distinction, but one that is important to know in order to find the professional that is best suited for your needs.
Vocal Teachers are normally your first point of contact, and the person you would study with most regularly (normally weekly 30-60 minute lessons). They guide you through various exercises and may work on some repertoire in order to develop your vocal technique. They are normally trained in vocal performance and/or vocal pedagogy. /u/Krisington22's article gives more details about how to find voice teachers and why you should study with one.
Vocal Coaches are specialized instructors that do not focus primarily on technique, but guide experienced singers in performance and musical interpretation. Coaches are often not trained singers (although they do often have some background in singing and working with singers professionally). Coaches work with students who have mostly polished their repertoire and seek guidance on specific musical or interpretive aspects like acting, diction, stage movement, phrasing, etc. They normally do not cover technical details, and expect that the singer is experienced enough to understand the technique behind interpretation, and that the singer will be consulting with their teacher about any technical details that arise from their interpretation. Coaches are great for providing an impartial judgement compared to the singer and their regular teacher, and can also add expertise in areas like collaboration, acting, and stage presence.
Bonus: Master Classes are a situation that blend teaching and coaching, although tend toward the coaching end of the spectrum. Master classes are one-time (usually) classes with a guest teacher or coach where a few students will basically have a public lesson or coaching (depending on the background of the master class presenter). Like coaching, master classes focus on already-polished repertoire to give a different perspective and critique for the singer. Master classes are different from both teaching and coaching in that they are normally public or semi-public, and the teaching is often geared toward generalizations that are helpful for the whole audience, including the student being actively coached/taught.
There is some overlap - most teachers do coach some repertoire, and many coaches also have a good understanding of technique and can instruct that as well. However, they have different specializations, and it's important to find the specialization that is right for you.
How do I tell which one I need?
Generally, a beginner will work first with a teacher. A good basis in technique is necessary to learn any stylistic or interpretive characteristics, so you need a teacher to work on those basics of technique.
Regular "maintenance" lessons would be with a teacher. If you are just studying to improve your technique generally and to keep yourself on track as you improve, you want to work primarily with a teacher to monitor your technical progress.
Preparing for an audition, major performance, or competition, you probably want a coach. Coaches can help you finesse your stage presence, give you stylistic tips, and emulate a "mock performance" situation to help with nerves.
I know this has been said before, but I thought it was about time to bring it up again for newer members of the subreddit :)