It has to do with our immune systems. For example if you get a kidney transplant (and don't have an identical twin donor), the kidney will have cell markers on them that are used to recognize the tissue as "self". So the donor had a kidney with a certain set of proteins on the cells of all their tissues that their immune system can identify its own tissues as "self" and the immune cells will not attack them while in the donor's body (in a healthy person). The kidney recipient will have their own set of these proteins that do the same thing, however the donor and recipient have differences in these markers between them. Those differences cause the recipients immune system to recognize the donors kidney as not self but "foreign" so the immune system attacks. This is why we give immune suppressing drugs to transplant patients for life to stop the immune system from killing the organ. If you were lucky and had an identical twin, then these "self" markers would be identical too, and you could get a kidney from them and the immune system would not reject it and no immune suppressing drugs needed.

Instead of kidneys, lets say you injected tumor cells from the donor into a non identical twin recipient what happens? The vast majority of the times the immune system will recognize these as "not self but foreign" and immediately attack and kill them. This is why most of the time you can't exchange cancer cells with another person and give them cancer. Again, identical twins? Different situation. Their "self" markers are the same so cancer cells from one twin would not be attacked by the other twins immune system and could grow and give the recipient cancer. This simplified a bit gets the concept across.