I would say it depends on the range of influence the transplanted organ has on the rest of the body. A new set of eyes will have little influence beside their sensory function; a new esophagus will work mostly like the old one. More central organs will have bigger effects: a new liver will improve your fat and sugar metabolism; a new pancreas will be probably life-changing. A new, stronger heart may prevent ischemia in otherwise poorly-irrigated tissues, but will do little beyond that. A blood marrow transplant will essentially give you new blood, whose consequences will depend on the donor's genetic traits. Since blood interacts with the entire body, a new set of kidneys will have huge repercussions.