i could spend hours talking about fat injection but don't want to bore people here, so just the cliff notes, most of what people write is accurate, you will have some resorbtion and you will have some lumping, you will have to go back and get it evened out. get the best doctor, make sure he uses the right approach, easy to find out via medscape research how this is done.
if you compare it to normal filler, disadvantage is therefore it much much harder to reverse, you cant be as precise. advantages are it is much cheaper as filler can run into the 10.000s if you use the very doctor and have big areas, it lasts longer i.e. 36 vs 18 months, and best of all it looks more natural, you don't get that housewives of beverly hills looks as the viscosity is lower.
clarification question : you mean he will make the tip of the chin wider, or he will make the jaw wider, i assume it is the former.