APS-C is just the sensor size. Fuji cameras are both APS-C and mirrorless, as are some Sonys, Canons and Nikons. Olympus and Panasonic use micro four-thirds sensors, which are smaller and therefore in theory, don't offer as much detail (which isn't necessarily a problem depending on your needs). The big advantage of a bigger sensor with something like wildlife is that you'll tend to want to crop your pictures because you can't get close enough, so they let you do that without losing too much detail. You can still take perfectly good pictures though. I took this on a micro four-thirds sensor:
But that's the full size of the image. Perfect for instagram, but you probably wouldn't want to print it. You'll notice a bit of noise in the background, which would be reduced with a larger sensor. That's because I had to crop this image quite a bit to get the animal framed nicely. Also if you're shooting wildlife in a forest, there's often not a lot of light, so the bigger sensors will perform better there.
The other advantage of both micro four-thirds and mirrorless is size and weight. I was able to carry my camera and three lenses all in a pretty small shoulder bag. With my full frame camera, I can't fit the camera and one lens in the same bag.