Like amelialw said, teaching is usually the reference used for someone whom is still learning the technical facets of playing while coaching is an entire area where technical issues are not discussed - musical facets are. But coaching usually involves someone who is not the student's piano teacher.
Think of a football team. The players were not hired so they could learn how to play the sport; they were hired to play and aid in the team's winning potential. The coach is the person who manages what the individual players do to be collectively considered a 'team'. But he doesn't teach the players the rules of the game or how to play - it is assumed they already know these things. The coach drills the players on tactics, team plays, and physical conditioning/strengthening related to the game. The coach evaluates the team as a whole and if he is good coach, can harness their collective potential by evaluating how individual strengths and weaknesses can be balanced within the team to maximize the teams winning potential.
Similarly in piano study, a coach is one who evaluates the student's current abilities and attempts to maximize his performance potential. This is the goal. The coach will then attempt to have the student realize certain aspects of his own performance and either attempt to enhance desireable qualities or get rid of bad ones so that his performance potential can be realized. But the coach does not teach how to play the piano - it is assumed he already plays with near master abilities1.
But coaching a football team and a pianist/musician is different in that one involves more than one individual2. The other isn't much different from a good teacher.
The great challenge of piano teachers are that they must teach the mechanics of piano playing while teaching the student to understand music so their students have a musical goal toward which they learn their technical skills. These two cannot be separated (though historically even up to now, it has been attempted to disastrous results). Thus, if the term coach is used to describe a person who deals with the purely musical issues, then a coach cannot be used to also mean a teacher. But a teacher does mean that he also coaches along with teaching.
1 Maximizing potential also assumes no limit to the service of making music regardless of any technical issues. But this is rarely the case. A coach cannot coach effectively if the students cannot play their instruments with absolute surety just as no coach can show a human to fly with great meaning - the human must already know how to use his wings.
2 In coaching a chamber ensemble, the coach does not teach the individuals how to play their instruments, they attempt to maximize their performance potential as a group. This can involve issues of ensemble playing, voicing, balance of tone, stage presence, etc.