Very big difference.
The leaves change color and drop off because the plant is going into a dormant state. The changing of the color is because the leaves have been cut off from a nutrient supply and die. The colors they turn are a result of decay and is incidental.
Animals changing color to better suit their environment is a whole different matter. These colors are deliberate so as to better blend in with the organism's environment. This changing of color goes down to the genetic level. New proteins must be coded in order for the change to take place from one color fur to the other. It's a completely different process.