The difference between vorticity and helicity
is that "vorticity" is a property of a fluid flow related to local angular rotation; defined as the curl of the flow’s velocity field and "helicity" is the quality of being helical.
vorticity

helicity

Noun
 (mathematics, fluid dynamics) A property of a fluid flow related to local angular rotation; defined as the curl of the flow’s velocity field.
Exemple
 It might sound puzzling that the ageostrophic geopotential has no impact on the evolution of the vorticity of a quasigeostrophic disturbance and yet it affects the perturbation kinetic energetics .
 This results from the vorticity component of shear, whereas elongated cells would further be affected by the rate of strain component.
 An extreme version of vorticity is a vortex. The vortex is a spinning, cyclonic mass of fluid, which can be observed in the rotation of water going down a drain, as well as in smoke rings, tornados and hurricanes.

Noun
 The quality of being helical.
 (physics, fluid mechanics, electrodynamics) Any of certain measures of the extent to which vortex lines (in fluid flow) or field lines (in a magnetic or electromagnetic field) kink and twist and/or link and coil around one another.
 (physics, quantum mechanics) The quantized spin component of a moving particle along the direction of its motion.
Examples
 The conjecture made in Tsinober and Levich was that coherent structures should possess a significant coherent helicity.
 The helicity associated with the writhing number is obtained by observing that a torus with twist number ±1 may be distorted into a figure8 configuration which appears untwisted. […] Note that at each step the total helicity, consisting of the sum of twist, kink, and link helicities, is conserved.
 Thus the total helicity H equals the sum of the entries in a matrix Hij. If N is large then there will be many more mutual helicity terms. In this case ignoring the self helicities may only give a small error.
 Now conservation of helicity is maintained: the writhe of the loops in the second panel is compensated for by the opposite sign of the twist helicity along the loops.
 To understand this name, note that the helicity is, roughly speaking, the spin in the direction of motion.
 The experiment was particularly ingenious because, as the neutrino hardly interacts with matter, some way of indirectly measuring its helicity had to be devised.
 Interpreted in terms of helicity, this means that if the incident electron has positive helicity, then the outgoing electron has positive helicity for forward scattering, and negative helicity for backward scattering.
