To be fair, there is sort of a qualitative difference between the two: the spirals of spiral galaxies are density waves moving independently from the orbiting s

To be fair, there is sort of a qualitative difference between the two: the spirals of spiral galaxies are density waves moving independently from the orbiting stars themselves. When it comes to fractals, there is a much deeper and fundamendal distinction that can be made, though: the difference between connected and disconnected sets. The Mandelbrot set is connected (and indeed simply connected), no matter how much some of its regions might look like made of disjoint components. There can always be found "tendrils" connecting everything to the main bulb. On the other hand, Julia sets are either connected (precisely those that correspond to points in the Mandelbrot set) or disconnected (in which case they are "dust" made of uncountably infinite disjoint points).