How are they more ambiguous than fractals? The key structure is a shape that can be rotated onto a subset of itself.
I guess with common fractals, you camnoly scale an object onto a subset of itself. The magic of BT is that the ball-shape allows a rotation that looks like a scaling.
A fractal tends to be kind of contiguous - for most points in the fractal there's a neighbourhood of those points that's also in the fractal, and similarly for points not in the fractal. These are "dust sets", dense everywhere in the sphere and their complements also dense.
Fractals generally have a formula describing them, i.e. they are computable. Nonmeasurable sets are necessarily non-computable. This is why you will never see an accurate picture of the B-T sets, because they are not defined uniquely by any formula or algorithm (which is connected to the fact that the axiom of choice is needed to even prove these sets exist).
I guess with common fractals, you camnoly scale an object onto a subset of itself. The magic of BT is that the ball-shape allows a rotation that looks like a scaling.