Fractal curvature might be the answer. In differential or convex geometry, you need curvature to classify sets up to isometry. So it seems natural to introduce curvature for "fractals" in an attempt to get a finer geometric description. This has been done for mostly self-similar fractals by Winter, Zähle, Rataj, Kombrink, and me (Bohl, formerly Rothe). The full generalization to self-conformal sets is my upcoming PhD thesis.

Philosophically, and literally in differential geometry, curvature takes the second derivative of "coordinates" into account. In contrast, the Hausdorff and packing measures, most other dimensions, Minkowski content (=lacunarity), and surface content are only sensitive to the first derivative. Topological entropy is related to Gibbs / equilibrum measures if you have some kind of iterated function system, and these measures also belong to first order geometry.