Single Hose - draws indoor air for both front cold air output as well as exhaust air (used to cool compressor).
PROS:
1) Fresh air is exchanged within the room.
2) Higher energy efficiency.
CONS:
1) Percentage of the air taken in is used to cool the condenser. As this air is exhaust out, it creates a negative air pressure. Air is continuously seeping in from adjacent rooms to replace the exhausted air.
2) Slower initial cool down.

Dual Hose - draws indoor air for front cold air output and draws outdoor air for exhaust air (used to cool compressor).
PROS:
1) If outside temperature is lower than indoor temperature, compressor can be cooled down faster and produce colder front air output.
2) No negative pressure created within the room
3) Initial cool down of room is quicker than single hose.
CONS:
1) Same air circulating within the room, which will decrease in oxygen and increase in carbon dioxide.
2) If outside temperature is high (>90°F) or when humidity is high (>80%), compressor cannot be cooled and results is increased temperature of front air output. In such cases, the second hose should not used to prevent the compressor from overheating.
3) Uses two internal fans and consumes more energy due to added hose.