I understand that it is possible to hire a business coach, but in what situations is it more appropriate to work with a mentor than with a business coach? And do some entrepreneurs work with both business coaches and mentors?
The major difference between a coach and a mentor is the purpose that each serves.
The mentor talks and gives advice mostly on principles and underlying philosophy about doing something. Even though, he/she has an established path of success there are no similarities, only resemblances. The mentor will make you aware of the challenges and opportunities that lie ahead, as well as of some ways of dealing with them.
The coach is your accountability partner. It's the one that will push you to move forward. The coach will make look straight at your b/s and assist you in overcoming it. The coach doesn't know what lies ahead but he will join you in your journey of exploration.
If you have both of them in your team, you are well on your way to success... be aware though, not to ask from either of them something that is not appropriate - (for eg. advice from the coach and accountability from the mentor!)