What is the difference between Nottale's "scale relativity", and the ordinary concept of scale invariance e.g. that appears in conformal field theory?
The point is that these two concepts are totally different and they have only superficial resemblances.
- Scale invariance is an extension of the Poincaré group (and a subgroup of the conformal group) implying that there is no scale at all in the theory. In particular there are no masses, no distances and no energies. There is no way to distinguish between two energy (or length) scales because a scale transformation an take you from one to the other.
- In scale relativity the Planck length plays the same role as the speed of light. Hence it provides a preferred scale to the theory. Moreover the theory looks scale invariant in some sense because the spacetime becomes fractal, but this is just a consequence of the ways length are "added": the usual addition does not hold anymore, in the same sense that you cannot add speeds in special relativity. But the point is that there is still a meaning to the question "what is the energy of my particle", and particles can have masses, and so on. This is like in special relativity: there is an "infinite" distance before reaching c, and you can always try to get closer by accelerating, but the value of the speed is physical.
Note that the doubly special relativity is another model with the Planck length being the minimal length, but the spacetime structure is different.
Nottale's theory assumes non-differentiability of space-time. The main equations work as corrections for ordinary differential equations.