According to Dictionary.com...
Consultant: n. 1. a person who gives professional or expert advice. Syn: advisor, authority, counsel, expert, freelancer, guide, master, maven, mentor.
Coach: n. 5. a person who trains an athlete or a team of athletes 12. to give instruction or advice to in the capacity of a coach; instruct. Syn: drill instructor, educator, mentor, physical education instructor, skipper, teacher, trainer, tutor.
For some reason, I find only a few vague differences in these definitions. The distinctions between a consultant and coach seem a little too general, especially in a case such as this. Technically, because of shared synonyms (i.e. "mentor") and job descriptions (i.e. to advise...), a coach might act as a consultant and vice-versa. It seems to me that the only difference in these two positions would be their given titles. Hmmm...
