Question: What is the difference between a nuclear melt down and a melt through? What are the implications?

There is a bit of a difference in that a melt through is where the fuel melts through the lower core plate and through the lower head of the pressure vessel into the primary containment.  It makes clean up a bit more challenging.  There may be some additional releases to the environment.  However, there is nothing necessarily catastrophic about this problem.  It's a bit like the difference in a turbine explosion of having all the blades contained, or a few actually being ejected past the engine containment.  Either way, your day is going to suck.   This is not the nuclear equivalent of a disk explosion.

There is actually some benefit to having some melt through because the fuel is now spread out and you have a larger surface to volume ratio allowing more cooling potential of radio nuclide decay heat.   The fuel that falls to the bottom of containment is also under water and in contact with a huge heat sink.  Part of the trick to cooling the fuel is lots of wetted surface area.

See the diagram below.  The reactor pressure vessel (brown) is inside the primary containment (yellow).  Leaking out of the reactor vessel does not mean being released to the environment.  Assuming the primary containment is still substantially or completely intact, melting through the reactor vessel into the drywell should not cause much additional release to the environment.

The primary containment is protected from steam overpressure by venting into the Torus which is partially filled with cold water.  The steam released from the RV into containment during an accident is vented underwater in the torus and condenses reducing its volume and controlling pressure in the primary containment.