Why do so many smokers die early? Effect on heart or cancer?

Many effects. Smoking and tobacco use in general, is a major etiologic factor in many cancers, causes lung disease, copd, in addition to lung cancer, accelerated atherosclerosis leading to heart attacks, strokes, lower limb gangrene. Smokers in general die ten years sooner than non-smokers. Tobacco use is the largest preventable act in promoting health and wellness.
Chose your poison. I'd suggest since cardiovascular disease is the leading cause of death in this country, that it may kill more smokers than cancer, especially considering the extra burden on the heart from smoking induced copd. Smoking also increases the risk of head+neck, bladder, and breast cancer in addition to lung cancer, so it would be very close.
Both. Along with consequences of their other addictions.