What's the difference?

There are no black and white answers to this question but I can share some [very simplified] examples and the scope that I work within. Note that often [depending on experience and qualifications] someone can do various roles in their workspace. If you’re after some objective information on what Coaching [in Australia] is about, and on how to choose a Coach, click this link to Choice Magazine.
The Hobajing focus is Coach, Train and Process. That is, I help look at what you have now, where you want to be and figure out the stuff in between, sometimes instructing you in specific skills. As a Coach and Trainer, I am not interested in the past. I am not saying that it is not important, but it’s not for me to determine if it’s relevant to your situation. Also, unless stated, information below is based on Australian requirements.

Professional Australian Requirements

Consultant [Life] Coach Trainer Counsellor
.. Dependant on area of expertise, for instance I would expect that a Consulting Psychologist, would also be a registered Psychologist. .. None required .. Dependant on area of expertise, for example to teach at TAFE, a Cert IV in Training and Assessment plus five years of recent experience in the subject being taught .. Minimum training for registration is either a Graduate Diploma or three-year Bachelor’s Degree
For More Information
Search Google for the specific Job Title, e.g. Psychotherapist, Book Keeper, Business Manager International Coach Federation or International Association of Coaching
Australian Institute of Training and Development Australian Counselling Association
Discipline Application: Jerry wants to lose weight
Consultant Coach Trainer Counsellor
.. provides tools & methods of weight loss .. helps research, work out management plan, resources available; inspire / motivate towards goal .. instructs on nutrition, cooking meals, exercise .. diagnoses issues that may affect weightloss, such as emotional eating, associated childhood trauma etc
Discipline Application: Overcome Jackie’s Project challenges
Consultant Coach Trainer Counsellor
.. shares similar experience and tools that improved situation .. defines group target, generates process from group workshop .. equips group with problem solving techniques and budgetary skills .. investigates source of anxiety and advises best course of treatment

The following websites contain helpful discussion on the various disciplines.