Why are Tasmanian Devils important? Because they occupy a very important niche in the ecological environment in which they live. Without them, other species would expand and/or take over the niche that the Tasmanian Devils occupied.
“The devil hunts at night, making it nocturnal. They hunt snakes, birds, lizards, rodents and some types of wallabies. It also tries to eat animals bigger than itself. The devil can be helpful by controlling the level of feral cats and other unwanted animals. They also eat dead animals and sometimes the older devils eat the younger ones. This is why some people think that young devil’s are good climbers, so they can get away. The devil’s have very strong jaws for crunching the bones of its prey. It can bite as hard as a big dog. The devil will eat all of its prey from fur to bones. The devil’s eyesight is suited for night, but whether in daylight or at night the devil’s have trouble seeing things that don’t move, so it is very hard at night to find dead animals, they have to catch then eat its prey. If you are bitten by a devil it can prove fatal. It is advised to seek medical assistance if bitten.“ (Tasmanian Devils - Prey)
As you can see,without the Tasmanian Devil, its primary prey could easily multiply significantly, and would disrupt the rest of the food chain and potentially the entire environment. That has happened here in the US with deer, which have proliferated dramatically without wolves and other prime predators. The result has been very dramatic changes in the environments where they live.