Antigens on the surfaces of cells in animal organs are drastically different from human antigens, rendering them very easily recognized and targeted by the immune system, leading to organ necrosis and rejection. This rejection may be mitigated by immunosuppressants but this would mean that the patient will have to take immunosuppressant drugs for the rest of their lives and live with an increased risk for infections due to immunosuppression.
Physiological differences between species
Physiological differences between humans and the animal species means that the animal organ may not be designed to perform the same function that it does in humans or it performs the function to a different extent.
Certain diseases can cross the species barrier and infect the patient if the animal organ is not appropriately chosen.
Certain patients may not wish to have animal organ transplants for religious, animal rights or other psychological concerns.
Animal organs, especially those from domesticated species (sheep, pig, dog, etc.) are easily accessible as these animals are easy to breed and their anatomies are well understood.
While no cost can be assigned to a human organ, an animal organ can be bought or sold at a price proportional to the animal's economic value.
Animal transplantation is more justifiable than human organ trade.