Question

Why can animals live on a diet that is only protein for long periods of time, but not on a diet that's only fat?

Answer 1

That is because proteins provide them energy, nitrogen, and direct units to make more proteins, they contain Carbon, Nitrogen, Oxygen, Hydrogen and even Sulfur. On the other hand, fats are much better energy source than proteins but they only contain Carbon and Hydrogen, so an organism can do nothing more with fats than energy obtaining and storage.

In simple words, proteins provide many different elements required by the body, and fats don't. Nitrogen, specially, is a crucial element for life that fats do not contain.