Apoptosis is programmed cell death undertaken by the body as part of a normal biological process in multicellular organisms. This is accomplished through biochemical reactions. This is an internal process.
Whereas apoptosis is a normal process, necrosis is an abnormal process. Necrosis means death of cells and or tissues or organs. Remember, cells make up tissues which make up organs. If something is necrotic, it is dead. Necrosis occurs in living tissues either from disease or trauma. Necrosis has an external etiology. A good example of necrosis is when a diabetic injures a toe, the toe becomes infected, the toe starts to die (become necrotic) because the diabetic tissue can't heal normally.
Apoptosis is programmed cell death. That means it is a "suicide" signal from the nucleus that tells cells to die and a specific time under specific conditions. (As a fetus your fingers and toes were webbed, apoptosis told the unneeded cells between you fingers to go away before you were born).
Necrosis is the death and rotting of cells usually as a part of infection, disease or injury. Some bacteria cause necrotizing fasciatis -- a bacterial infection where cells dies and rot very quickly and without treatment fatally for the patient. Some venoms in snakes and spider can cause necrosis.