The difference between Loss and Release is pretty simple, on the surface. In either case the person is dead and it seems like everyone forgets are them. In the case of Release, the forgetting takes longer. In the case of Loss, the person has died an accidental death and things are so tightly controlled that this is quite unusual. They perform a ceremony where they repeat the person’s name until they pretty much forget him or her, and then give a newborn the person’s name.
Release is the death penalty or euthanasia. When a person breaks a rule three times or commits one serious infraction, he or she is given lethal injection. When a person is old, he or she is euthanized. Babies are also euthanized when they don’t live up to expectations. Release is also sad, except in the case of the Old because they are considered to have lived long enough lives.
In The Giver, "loss" is when someone dies a natural death. "Release" is when a person is put to death for some reason.