Importance of MHV amplitudes Why are MHV amplitudes so important?  How/where are they used and why do people keep trying to rederive them in many different ways?
 A: MHV amplitudes are not really any more important than next-to maximal helicity violating amplitudes ($NMHV$) or $N^kMHV$ amplitudes. You need all of them to compute a general scattering amplitude.
Basically, scattering amplitudes for non-Abelian Yang Mills theories are very complicated to compute for more than 4 particles, so people work on formulating easier ways to do the computations. The first step is usually to strip off the color dependence of the amplitude, so that the object you really need to compute is the color ordered amplitude $\mathcal{A}(\pm,\pm,...\pm)$. These objects get multiplied by traces of color matrices at the end of the calculation. 
At tree level, if all particles have the same helicity or only 1 helicity is different, the color ordered amplitude vanishes. So the MHV amplitude is the first non-vanishing color ordered amplitude. For $2 \to 2$ scattering you only need the MHV, since you can have at most 2 $+$ and 2 $-$ helicities. But if you were to compute say $3 \to 3$ scattering, you would also need something like $\mathcal{A}(+++---)$ which is not MHV.
MHV probably gets a lot of attention because the formulas are extremely simple (see: Parke-Taylor formula). Moreover, the discovery of the BCFW recursion relations allow you to generate higher order amplitudes from the lower order ones. In general, people are deriving new methods to calculate the amplitudes, and they can check their methods by matching onto the MHV amplitudes which have a simple form. But to compute a general amplitude you need more than just MHV.
Hope this helps.
