Perhaps the shortest answer (from my elementary understanding of representation theory) would be that the differences between the types of spinors you asked for lie primarily in terms of the representations of the rotation group under which they transform: If the spinors are symbolised by ψ, then the transformation rule:
Ψ’ = M Ψ, where M is one of the matrix reps of the rotation group.
M = SL(2,C) for 2-component Weyl (and other relativistic, e.g. Lorentz) spinors, which obey the Weyl equation (the massless form of the Dirac equation)
M = SU(2) for the non-relativistic 2-component (Pauli) spinors, which obey the Schrodinger-Pauli equation – the non-relativistic but massive limit of the Dirac equation (I suspect their components, for normalised wave functions are each restricted to the unit circle in the complex plane.)
M = something much more general than either of the above for Cartan spinors in their most general form (possibly via SO(p,q) for general p, q?) as Cartan claimed his spinors are the most general mathematical form of spinors, and they deal with rotations in spaces of any number of dimensions. They should therefore range beyond 2-component objects (as do even Weyl spinors for e.g. 6-D space).
Although you profess disinterest in Dirac and Majorana spinors you might also like to refer to a comparable type (but much more expert) answer comparing Weyl with Dirac and Majorana spinors.
Meanwhile, as others can probably provide a better answer, I regard this as an opportunity to learn more by inviting corrections; otherwise, I might post a more detailed answer, with references, later.
UPDATE: Abstract of the ref given as this paper in Physics StackExch 381625 states:
"...The physical observables in Schrödinger–Pauli theory and Dirac theory are found, and the relationship between Dirac, Lorentz, Weyl, and Majorana spinors is made explicit." (behind a paywall).