The people who get them have usually suffered some kind of extremely traumatic injury. IIRC the first woman to have one was mauled by a dog, which literally ate her face.
Before face transplants, the only option was to take skin from elsewhere and use it to repair the face as best as possible, to he point where it looked awful but at better than having no skin. The upside was that because the skin came from their own body, rejection wasn't a big problem.
Face transplants were supposed to offer better functionality (ability to talk, eat and breathe more normally, as well as facial expressions) and look cosmetically a bit nicer.
There does seem to be some kind of issue with "external" transplants like this. I recall someone who had a whole hand transplanted, but wanted it removed. Even aside from issues like parts of the transplant not getting enough blood flow and dying, it seems that having part of someone else's body visibly grafted onto you (not an internal organ) comes with psychological challenges.