I wouldn't say there are results that can only be obtained through differintegration. It only happens that there are problems whose solutions look neater when we bring in the machinery of differintegrals.
Spanier and Oldham and Miller and Ross remain useful references on the applications of differintegration. The first reference has a chapter on how certain diffusion problems have a neater formulation when differintegrals are used. For the second reference, the application that jumped out at me was Abel's solution to the so-called tautochrone problem: finding the curve such that the time needed for a particle to descend from a given position to the bottom of the curve (assuming there is no friction) is independent of position.
Though Huygens and other mathematicians have already obtained this solution long before Abel, he decided to use an integral equation formulation that can then be solved with the help of differintegration. In particular, he arrived at the equation
which when reformulated as a differintegral is
I won't spoil the rest of the solution; I'd suggest that you read Miller and Ross if you're interested.