Twin spool engines are more fuel efficient than three spools.
See ICAO Engine Emissions Databank....
Aircraft Engine Emissions | Human and Environmental Issues | Safety Regulation
Note: Compare engines with similar thrusts.
For the 777 with the GE90, the sea level SFC is 0.324, at cruise it is 0.52.
For the 777 with the Trent 800, the sea level SFC is 0.35, at cruise it is 0.56.
For the 747-8 with the GEnx, the sea level SFC is 0.27, no figures yet for cruise.
GE90 SFC (SLS) 8.30 mg/N-s (cruise)
Trent 882 SFC (SLS) 15.66 mg/N-s (cruise)
Twin spool engines have a more stable airflow pattern since the airflow is being compessed all the way to the last stage of the HP compressor before it enters the diffuser section.
This makes the engine less surge prone than a three spool design.
In a three spool engine, there is a sudden interruption (slowing down) of the airflow in the void between the IP Compressor and the HP Compressor. When the compressed air leaves the last stage of blades of the IP compressor no more compression takes place until the first stage of the HP compressor blades. This airflow interruption between the two compressors in this uncompressed void makes the engine more surge prone.
Twin spool engines, (GE or Pratt) have less heat to dissipate than three spool (Rolls Royce) engines.
Three spool engines operate at a higher oil temperature when compared to two spool engines and the oil distribution is much more complex in three spool engines. This more complex oil distribution has given RR problems over the years, some call it oil hiding.
RB211 Series 335'
Trent 700 374'
Trent 800 375'
Trent 900 385'
Trent 1000 365'
CF6-50 320'
CF6-80C2 320'
PW4000 350'
GE90 270'
Internal engine cooling airflow is less complex in a twin spool engine than a three spool one.
For this reason triple spool engines emit more smoke during start-up.
Twin spools engines have lower gyroscopic moments resulting in less side loading of the pod/strut. (pod nod)
Twin spools light off and accelerate faster than three spools.
Compare the slow spool-up time of a RR Trent compared to a GE or Pratt engine in the following link.
YouTube - L-1011 N700TS Airline History Museum 1 ..........( this almost sounds like a hung start as it takes such a long time to start )
Most fighter aicraft engines use a twin spool design for faster throttle response.
Twin spools have lower maunfacturing costs due to a lower parts count.
Twin spools are less expensive to overhaul due to the face that they have only two concentric shafts, no third (Intermediate) compressor with it's associated compressor and stator blades as in a three spool design.
The RR three spool is more difficult and more labor intensive to manufacture beacuse of the nature of the concentricity of the drive shafts, support bearings and the fact that is has three distinct compression stages.
An Oxford University/Rolls Study from 4/9/02 - 9/30/03 document notes that 10% or more of RR engines fail the final passing out test due to imbalance. 11 Trent 500 production engines failed pass-off testing due
to abnormal vibrations. This indicates that there was a systemic vibration problem. A higher rejection rate due to vibration is detected when the engines are overhauled at the RR appointed agents.