What's keeping living species from rotting?
One can calculate that there are more non-human cells (mainly bacteria) inside human body than there are actual human cells. Why those bacteria don't eat us from within? Epithelium is pretty good barrier between outer world and internal systems. It is essentially one-cell thick layer that protects internal organs and infrastructure. In cases when epithelium breaks, bacteria can get into blood stream and start very bad infection, which can cause necrosis (effectively, rotting, tissue decay). Natural cell death is fought by natural cell proliferation so that there are no gaps in defense layers.

Also one should not forget about immune system that is looking for invaders or anything that should be where it is. In healthy person immune response will close gaps in epithelium and kill bacterial infections pretty quick.

Dead people, as well as dead plants, have their skin and epithelium, immune system destroyed by cell death or infection. Then bugs start to eat vessels, tasty nerve tissues, muscle and cartilages, blood cells filled with proteins and everything eatable.