Fame
Every once in a while, a particularly juicy story about animal rescue makes it way into mainstream media, and if there's a big oil spill or some major catastrophe that affects animals, the story will definitely get news coverage. In instances like these, wildlife rescue teams get interviewed.
In general, people like animals and they don't like seeing animals in danger. Exciting pictures of friendly people saving animals makes for good Internet clicking. So should you ever save a duckling by busting out a quacking ringtone, prepare for your phone to start ringing.
So if the stars align, and you manage to rescue just the right photogenic goat, or you're the guy who airlifts Garfield out of a tree, there's the slimmest of slim hopes that you could become Internet famous. For the vast majority of animal rescuers, though, you're never going to turn on the news and see yourself. Or if you do get media coverage, it will just be a short write-up in a local paper: "Cat rescued! Cat lady relieved!"
So yes, there's a bit more potential for fame than say, working at the 7-Eleven. But Robert Downey, Jr., you are not.