The type of fat does matter. The body first metabolizes carbs then protein,
and while the protein is being metabolized, the body starts breaking down
the fat and glycogen stores to convert it to fuel. Glycogen is a long
string of simple carbohydrates, which gets cleaved and then the carbs
(sugars) enter Glycolysis, which is the metabolic pathway by which we
metabolize sugars.
Fat, however, does not have the same chemical structure as sugars, but
miraculously, the body breaks it down into the same product as which
Glycolysis breaks down sugars. This product is called Acetyl-Co-A, which is
a very small molecule from which we ultimately derive a bit of our
energy. It of course, does this through a completely different metabolic
pathway called beta-oxidation.