Thermoplastics, in basic terms, are melt-processable plastics (materials that are processed with heat).  When enough heat is added to bring the temperature of the plastic above its melt point, the plastic liquefies (softens enough to be processed).  When the heat source is removed and the temperature of the plastic drops below its melt point, the plastic solidifies (or freezes) back into a glass-like solid.  This process can be repeated, with the plastic melting and solidifying as the temperature climbs above and drops below the melt temperature, respectively.  However, the material can be increasingly subject to deterioration in its molten state, so there is a practical limit to the number of times that this reprocessing can take place before material properties begin to suffer.  Many thermoplastic polymers are addition-type, capable of yielding very long molecular chain lengths (very high molecular weights).

Figure 1:  Gr/PEEK^TM helicopter cargo bay floor panel has toughness benefits thanks to its thermoplastic matrix.

Thermosets, again in basic terms, are materials that undergo a chemical reaction (cure) and transform from a liquid to a solid.  In its uncured form, the material has very small, unlinked molecules (known as monomers).  The addition of a second material (catalyst) and/or heat or some other activating influence will initiate the chemical reaction.  During this reaction the molecules cross-link and form significantly longer molecular chains, causing the material to solidify.  This change is permanent and irreversible.  Subsequently, exposure to high heat will cause the material to degrade, not melt.  This is because these materials typically degrade at a temperature below where it would be able to melt.

Figure 2:  Graphite/BMI compressor rotor uses a thermoset matrix to take advantage of its processability.

Each material has its place in the market.  In broad generalities, thermosets tend to have been around for a long time (have a well-established place in the market), frequently have lower raw material costs, and often provide easy wetting of reinforcing fiber and easy forming to final part geometries (are easier to process).    Thermoplastics tend to be tougher (less brittle), can have better chemical resistance, don't need refrigeration (as uncured thermosets frequently do), and can be more easily recycled and repaired.