What's the difference between helicity and chirality?
At first glance, chirality and helicity seem to have no relationship to each other. Helicity, as you said, is whether the spin is aligned or anti aligned with the momentum. Chirality is like your left hand versus your right hand. Its just a property that makes them different than each other, but in a way that is reversed through a mirror imaging - your left hand looks just like your right hand if you look at it in a mirror and vice-versa. If you do out the math though, you find out that they are linked. Helicity is not an inherent property of a particle because of relativity. Suppose you have some massive particle with spin. In one frame the momentum could be aligned with the spin, but you could just boost to a frame where the momentum was pointing the other direction (boost meaning looking from a frame moving with respect to the original frame). But if the particle is massless, it will travel at the speed of light, and so you can't boost past it. So you can't flip its helicity by changing frames. In this case, if it is "chiral right-handed", it will have right-handed helicity. If it is "chiral left-handed", it will have left-handed helicity. So chirality in the end has something to do with the natural helicity in the massless limit.
Note that chirality is not just a property of neutrinos. It is important for neutrinos because it is not known whether both chiralities exist. It is possible that only left-handed neutrinos (and only right-handed antineutrinos) exist.
Helicity is easy to define; chirality is more subtle.
The helicity of a particle is the normalized projection of the spin on the direction of momentum. If the spin is more along the same direction of the momentum than against it, then the helicity is positive; otherwise it is negative.
Chirality is to do with the way the particle's properties transform when they are described with respect to one inertial reference frame or another. The difference between right-handed and left-handed is like the difference between contravariant and covariant 4-vectors, but now we are talking about spinors. For a massless spin half particle, the spin and momentum can both be extracted from a single spinor. When one transforms from one frame to another, one should use the ordinary Lorentz transformation for a right-handed spinor, and the inverse Lorentz transformation for a left-handed spinor. Thus chirality is an intrinsic property of such a particle, but one whose influence is only revealed in this subtle way. It influences how the spinor enters into the Weyl equation, for example.
Massive spin-half particles such as electrons have their spin and momentum described by Dirac spinors which are made of two Weyl spinors, one of each chirality.
What distinguishes a neutrino (treated here as massless) from an anti-neutrino is primarily its chirality. But whenever just a single 2-component spinor describes both the momentum and the spin, one finds that the helicity for such a particle can only take one value (and for the antiparticle it takes the opposite value). Thus the helicity and the chirality then have the same value, but it does not mean they are the same thing.
When a given type of particle can only have one helicity, one has a situation that does not respect parity (mirror-reflection) symmetry. This is at the heart of the breaking of parity invarience by the weak force.