Technically plants die from cancer, is just that what we call cancer in plants is different that what we call cancer in animals.

Plants can stand better mutations than animals because they have a clonal structure and asexual reproduction. There are "bricks" in this structure called "leaves", "roots" or "stems between nodes", repeated again and again in the entire creature. Mutations lead often to stunted growing, thus health clones just overgrow and kill the damaged parts. They have an autoprune system also.