From the discussion in Quantum Diaries, it seems that helicity is a property of motion and chirality a property of shape (where, in the case of an elementary particle, this might be represented by something like the shape of a level surface of its wave function).
The language chosen by physicists is unfortunate as a helix is an object with fixed chirality but the chirality of the path of a “helical” motion depends on the relative motion of the medium in which it is traced.