The term "flamethrough" is unfamiliar. Sounds as if this might refer to the less elegantly designated tail pipe fire. When the crew has not followed the correct starting procedure for a jet engine, and fails to get ignition soon enough during the start-up, fuel can puddle in the combustion chambers (sort of like flooding the carburetor in older cars). This leads to spectacular amount of raw flames shooting out of the rear of the engine once ignition is finally achieved. A tail pipe flame can be dangerous for anyone standing too close, but rarely causes damage to the engine.
"Flame out" is when there is not sufficient fuel, or temperature, in the chambers to maintain combustion. In this case the engine fails to generate thrust. In flight, the crew can try for a restart using the ignitors in the combustion chamber. To help prevent possible flame outs from water ingestion in heavy rains, aircrews sometimes turn on the ignitors on to help ensure that fuel in the chamber burns. Ignitors are normally turned off at completion of the engine start-up procedure to increase their service life.
Then there are compressor stalls. Long story short, these occur when there is insufficient air flow at the jet engine intake, or when the compressor and turbine stages are not operating properly in relation to one another. This was a recurring problem for the early tri-jets (Lockheed 1011 Tri Star and 727) with jet air intakes on top of the fuselage at the rear of the aircraft. (This was addressed somewhat in the DC-10 design by mounting the engine well above the fuselage.) At high angles of attack, the body of the aircraft could interrupt airflow causing a compressor stall. Compressors stalls are followed by surges, experienced as a loud pop, or series of pops, when combustion resumes. Too many of these can damage the engine.
Finally, there are unrecoverable failures. These are most often due to loss of a turbine blade. These can be "contained" or "uncontained". Contained means that the Kevlar reinforced nacelle did its job and any debris that exited the engine did not do so through the nacelle and thus did not otherwise damage the aircraft. Uncontained means that debris leaves the engine nacelle, often damaging the wing or fuselage. The recent incident near Denver with debris falling into residential neighborhoods was a good example. In that case, the failure was accompanied by an engine fire fire that persisted well after the initial event, as shown on video taken by a passenger.
Perhaps Tapir Rider is familiar with "flamethrough".
Just now saw Res Ipsa's comment upthread about "flamethrough" vs. flame-out.
