The main advantage is in the ability to interchange lenses. You can switch from a telephoto, for its foreshortening effects and getting close-ups of more distant animals, to a wide-angle, faster lens for your panning shots at moving game (or kids).
Some cameras might give you most of that without the bulk and expense of extra lenses, plus the futz time spent deciding your strategy and changing them out. DSLRs are pretty bulky, and you might decide that you don't want the hassle of carrying it, and miss any shots altogether because you didn't have that little snapper in your coat pocket.
I have put off getting a DSLR for years, making do with a Canon point-and-shoot. I will get one, eventually, because I have three very cool lenses to use on one. But check out the pics that Elksong posted, taken with a Canon SX30IS. This camera appears to be the same size as my EOS film cameras, but all that lens capability is contained in the body.
My own little camera, a Canon G9, is much smaller, but does not have the same zoom capability as this other one. Shop around, learn about what you really need the camera to do, and you will be amazed at what is out there to pick from.
Killdeer