ASP stores fat in adipose tissue, but it's nowhere as effective as insulin. ASP secretion is controlled by chylomicrons, which are very short lived, and that makes it so that only a very limited amount of dietary fat can be stored as body fat.
>a very limited amount of dietary fat can be stored as body fat
In the complete absence of insulin, maybe, which is almost never the case in normal dietary conditions, since both protein and carbohydrate stimulate insulin secretion. But dietary fat is nonetheless the primary source of body fat; direct conversion of dietary carbohydrate to body fat is not a quantitatively significant process in humans except for in extreme conditions. See De novo lipogenesis in humans: metabolic and regulatory aspects. http://www.ncbi.nlm.nih.gov/pubmed/10365981
"Only when CHO energy intake exceeds TEE does DNL in liver or adipose tissue contribute significantly to the whole-body energy economy."