A heart transplant involves a patient receiving a donor heart. There are various surgeries, in addition to transplantation, which require opening of the chest - as skydnsr previously noted.
Open-heart surgery is any heart surgery in which the chest is opened up and the heart is exposed. Procedures like coronary artery bypass and valve replacement can be done in open-heart surgery.
Heart transplantation could, I guess, be referred to as open-heart surgery, because it does involve opening the chest and exposing the heart, but I think heart transplanation is usually just called heart transplantation. When people talk about open-heart surgery, they usually are referring to some type of procedure short of heart transplantation.