Lenny is confused, I think. They're closely related, pretty much the same thing. Light front dynamics is pretty much the same approach as "light cone gauge" except that "light front dynamics" etc. is used by people close to nuclear physics and "light cone gauge" is used by the stringy and nearby researchers. In both cases, we want to slice the spacetime along light-like slices similar to $ct+z=0$ which we call "one moment"; that's the limit of the slices in the rest frame of infinitely boosted observers (in the $z$ direction in this case). For gauge fields, we also want to impose some light-cone gauge for the components like $A_t+A_z=0$.
Both of them are really the limit of the "infinite momentum frame" for the momentum going really to infinity, up to some conventional rescaling of quantities by the infinite momentum, and whenever one may discuss things in the "infinite momentum frame", it is possible to actually take the equations to the limit and rewrite all the calculations in the light cone gauge. In the light cone gauge, all the components etc. are actually finite, like the $P^\pm$ components of the momentum. In the infinite momentum frame, the energy and one component of the momentum are... infinite but the scaling with the infinite number is universal and may be dropped, and that's how we get the finite light-cone-gauge formulae.
These issues were a source of considerable confusion e.g. in the Banks-Fischler-Shenker-Susskind paper founding Matrix theory that used the terminology of the "infinite momentum frame". People who realized that this was nothing else than a contrived way to talk about the light cone gauge, like myself, have been presenting all the results as results in the light cone gauge – which is discretized in the Matrix theory case. Lenny (who was inexperienced in the modern machinery of the light cone gauge although he was a co-father of IMF of a sort) rediscovered these well-known things many months later and made a big deal but there has never been any real deal about it. The infinite momentum frame is just an outdated, awkward way to use the same approach as the light front dynamics.
If someone needs a reference stating that they're equivalent, take e.g.
It says, on page 5, "It [light front dynamics] is also equivalent to the usual equal time formalism in the inﬁnite momentum frame".