Pongo;
Not sure exactly either as to their effect on in-flight performance.
As to the compressors. Two Stage implies that there are basicaly two air compressors (blowers / air pumps). They are physically connected in series. The discharge of the first provides the suction to the second. This provides a two step change in the boost pressure.
Uses for this type of compressor (non-aircraft), generally appear in applications where a high flow rate is desired. The horspower requirements are more efficient, than a single high pressure change across a single stage compressor.
I have a two 7 stage air blowers running downstairs.
The compressor has two main requirements for torque or horsepower (supplied by the engine). They are to compress (raise the pressure) and to pump (move)the air.
I suspect that the two stage blower would be marginally more efficient concerning Hp use, but the additional weight and complexity might be an offset vs a single stage compressor.
Truth be known, I wonder if it was actually a "Two Speed Two Stage" compressor, the name being truncated for common usage.
Interesting discussing fellas... Thanks!

------------------
Mino
The Wrecking Crew