A Singing or Voice Teacher will teach you the technique of singing - beginning with proper breathing and moving on to the technical aspects of voice production to help you:
a) find your voice if you are a beginner, and b) maintain it in a healthy condition if you are a professional. They will also help solve technical issues and give some musical guidance.
Once you feel comfortable with your voice production, you may decide you want to sing in public or present yourself for an audition. This is where a Vocal Coach steps in.

A Vocal Coach will help you perform your song or aria -
Most singers get to a stage where they have learned the notes and can sing them well enough but need inspiration in one area or another. It may be breath control or pronunciation, or they feel they want more sparkle to their performance, or perhaps a favourite aria has grown stale over the years and needs a new injection of life. Many singers like to have a coaching session simply to help them establish a new aria confidently in their repertoire.

A vocal coach will cover all aspects of performance art - in particular:

Interpretation - starting with the meaning of the whole song, then of individual words and how you wish to portray them;
Language - whether you are singing in your own language, a foreign one or a dialect - correct pronunciation is vital for the audience to understand you;
Expression - how do you colour the words with your singing voice…. loud, soft, fast, slow? … with what emotional charge? What indications has the composer left you? How can you get across to your audience what you wish to express about your aria using your sound alone? Does your face reflect the words and feelings you are singing about?;
Musical Knowledge - a vocal coach will have experience of the many different classical musical styles throughout history as well as contemporary genres like Musical Theatre, Rock, Pop & Jazz.
They can help you understand phrasing, rhythm, tempo, ornamentation and tradition appropriate to your song;
Vocal Experience - a vocal coach will also have knowledge of voice craft so they can alert you to good breathing options, vowel placement and ways of getting round difficult technical passages;
Presentation - a final, crucial element to performance is your mental and physical presentation - e.g. posture, demeanour and confidence, not just while you are singing but from the moment you walk on stage, together with best practice in various situations like auditions.