There is probably an old capacitor in the electronics which dried out over the time and now takes a long time to gain the necessary power load. If you are familiar with electronics you can check yourself for capacitors which looks bloated or otherwise funny and replace them.

Think I'll chime in here and add my two cents to these other good answers. There are two components to the original question:

1. How much less power does will the 3 phase A/C compressor unit use?
2. Longevity of three-phase motor versus single phase

Power

Motor Efficiency = Power Output / Power Input

It takes a certain amount of power to run the compressor regardless of single or three phase power... this is the power output. The power input is what you are trying to minimize, thus you want to increase efficiency as much as possible. The trouble is this metric (efficiency) is a hidden combination the components inside the compressor unit (compressor, fans). It is not necessarily true (but it usually is true) that three phase is more efficient than single phase.

I would look at the SEER rating which 'automatically' includes the motor efficiency for a better indicator of "cost to cool".

Longevity and Reliability

Three-phase motors and compressors are generally more reliable than their single-phase couterparts. But like anything else, there is more to it than that one attribute. Using Philps' automotive engine example, a 4 cylinder Honda might be more reliable that that old smallblock V8.

Vibration

Certainly less in the three-phase motor. And generally less vibration equals greater reliability.

Motor Starting

Three-phase motors have lots of starting torque and don't (typically) require any special circuitry to start (capacitors, centrifugal switches). And of course fewer components means fewer things to go wrong.

Cost

Is disappointing that the three-phase equipment is more expensive. Almost always, three-phase motors are actually cheaper than an equivalent single-phase motor.