Which are the advantages of low pressure levels during flight? (i.e during cruise flight in a commercial aircraft)
You want the lowest pressure that can be tolerated by most of the population for the purpose of maintaining adequate blood oxygen levels, which is generally considered to be 8000 ft above sea level. At 8000 ft, someone with degraded lung function, like a smoker, is effectively at 10000 ft plus and is already at the threshold of having marginal blood oxygen for peak brain function.  

At 40000 ft this requires a pressure differential of a little over 8 psi between the interior and exterior and you build the pressure hull to that and no more (it's a certification requirement).  As per min's comment this is partly to keep from having to make the pressure hull any stronger than it needs to be (every pound over the minimum is non-revenue ballast) and also to minimize engine bleed demand, which can be thought of as lost energy being used to keep the balloon of the pressure hull inflated instead of making thrust.  So, bottom line is it would be better for the airplane, both structurally and from an energy use standpoint, to operate with lower pressure differential and higher cabin altitude, but now the crew would have to start using supplemental oxygen and more people in the back would be getting sick.

The 787, because of its carbon fuselage, is able to go with a higher differential structurally, and its electric-pump pressurization system is a bit more efficient than taking bleed air directly, so Boeing had the option to certify to a higher differential pressure, 9 psi, which gives a 6000 ft cabin at service ceiling.  They could have used 8 psi and just shaved off more weight and used slightly lighter air pumps, but instead they increased the differential.  This was done mostly as a cabin comfort sales feature and was thought of as a bigger sales benefit than going with the weight saving route.