In tropical rain forest, the animals will have to deal with higher temperatures and greater rainfall. One example of an adaptation they might need is a thin, water-proof coat. This will prevent heat from becoming trapped inside their fur/hair/feathers, and the water-absorbing coating will allow water to stick to the inside of the coat without being absorbed, where the water will absorb the heat instead of themselves, and will cool the through evaporative cooling (heat release through evaporation)
In a deciduous forest, there is much colder temperatures, and significantly less rainfall. These animals would likely want a thicker water-proof coat. This is because the thicker coat would allow heat to become trapped, keeping them warm in the winter, and the water-proof coating prevents water from freezing in their fur, making them colder.