Comparing 5.56x45mm, 6mm PPC, and Undercharged 6.5x47mm Lapua
Assuming 14.5" Barrel

Tony Williams has published a provocative and compelling article, The Case for a General-Purpose Rifle and Machine Gun Cartridge (GPC).  In it, he outlined the need and advantages for an intermediate-caliber general-purpose cartridge which could fulfill the roles currently filled by 5.56x45mm and 7.62x51mm in infantry rifles and machine guns.
His article inspired me to examine a variety of already-existing cartridges and projectiles which might approach his GPC criteria.  I have no conclusions, but this article summarizes the state of my investigations so far.
Two particularly promising cartridges are the 6mm PPC and 6.5x47mm Lapua.
The 6mm PPC is a bit underpowered for Williams' criteria, but generates moderately more muzzle energy than 5.56x45mm, and provides sufficient headspace for the kind of long, pointed ogival projectile shapes which yield high ballistic coefficients.
The 6.5x47mm Lapua brackets Williams' criteria from the other direction: it is slightly too powerful to be used in an intermediate-power infantry rifle.  To compensate for this, it is assumed that a 10% reduction in charge will result in a 10% reduction in muzzle energy, which puts it exactly in line with Williams' critiera.  It, too, provides sufficient headspace for very-low-drag projectiles with long ogival shapes.
Hornady's A-Max line of projectiles provide the mass and ballistic coefficient (G1) parameters for the 6mm PPC and 6.5x47mm projections.  To achieve the long-range potential of the GPC, a high ballistic coefficient is more important than a high velocity or kinetic energy at the muzzle.  The 5.56x45mm generates a high muzzle velocity, and the 7.62x51mm generates a high muzzle energy, but both suffer from low ballistic coefficients because neither provides sufficient headspace for an optimally-shaped projectile.  A-Max is not the absolute market leader in ballistic coefficient, but it was selected because its ballistic coefficients are very good, and it is available in a variety of calibers, at the desired projectile masses, and with a composition given to good terminal effects.
To make the comparisons equitable, reported cartridge performances were adjusted to fit those expected when fired from a 14.5" long barrel (as seen on the United States Army's M4 infantry rifle).  This was achieved by multiplying the published 6mm PPC / 24" barrel muzzle energy by 0.886, and multiplying the published 6.5x47mm / 20" barrel muzzle energy by 0.930.
As an exception to this, the second 7.62x51mm NATO performance projection reflects use of a full-length 24" barrel (per United States Army's M24 rifle).
Projections of performance over range were generated via the Renegade Rifle Ballistics web page.  Included here are graphs of projectile energy vs distance and velocity vs distance.  Click on a graph to view the full parameters and results for that cartridge.
CartridgeVelocity vs DistanceEnergy vs Distance
5.56x45mm NATO, 55-grain FMJ, bc 0.189
5.56x45mm NATO, 70-grain Semi-Spitzer, bc 0.214
6mm PPC, 70-grain A-Max, bc 0.307
6.5x47mm Lapua, 123-grain A-Max, bc 0.510, 90% charge
7.62x51mm NATO, 149-grain M80 (BT-FMJ), bc 0.393, 14.5" barrel
7.62x51mm NATO, 149-grain M80 (BT-FMJ), bc 0.393, 24" barrel
Highlights of comparisons:
The 6mm PPC 70-grain A-Max velocity exceeds that of 5.56mm 55-grain FMJ after about 50 yards, and that of 5.56mm 70-grain semispitzer at the muzzle.  It exceeds the velocity of 7.62mm 149-grain / 14.5" barrel at all ranges, and exceeds the velocity of 7.62mm 149-grain / 20" barrel up to about 800 yards, beyond which the M80 retains slightly more velocity.
The 6mm PPC 70-grain A-Max kinetic energy exceeds that of 5.56mm 55-grain FMJ and 5.56mm 70-grain semispitzer by a strong margin at all ranges, and is outperformed handily by 7.62mm 149-grain by a factor of 1.5x at short ranges to 2.0x at long ranges.
The 6.5x47mm Lapua 123-grain A-Max velocity exceeds that of 5.56mm 55-grain FMJ after about 150 yards, and that of 5.56mm 70-grain semispitzer after about 50 yards.  It exceeds the velocity of 7.62mm 149-grain from a 14.5" barrel at all ranges (by about 3% at 200 yards, to 15% at 1200 yards), and starts outperforming 7.62mm 149-grain from a 24" barrel around 400 yards.
The 6.5x47mm Lapua 123-grain A-Max kinetic energy exceeds that of 5.56mm (all bullet masses) at all ranges to an overwhelming degree (it retains more kinetic energy at 1200 yards than 5.56mm develops at the muzzle), but is considerably outmatched by the 7.62mm 149-grain from a 14.5" barrel until around 500 yards, beyond which point the Lapua retains slightly more kinetic energy.  The kinetic energy developed by 7.62mm 149-grain from a 20" barrel outperforms 6.5x47mm Lapua by a large margin until about 800 yards, beyond which the two retain essentially equivalent kinetic energy.
Further considerations:
A key element not compared here is weight.  The slightly higher mass of 6mm PPC and (especially) 6.5x47mm Lapua ammunition adds to the rifleman's already formidible burden.  I am investigating the weights of these cartridges, and will draw up a comparison of the weights of typical ammunition load-outs with each of these ammunition types.
It is also worth noting that A-Max bullets are somewhat more expensive (at about $0.35 each) than 5.56mm FMJ.  Whether their performance is worth this expense is complicated, and hopefully will be the subject of educated debate.
Discusson on the GPC topic is ongoing in the TankNet forums.
See also the followup article: Exploration of General Purpose Cartridge Concept, Take Two