I see. Stopping power in what sense though?
So 7.62 mm can be used in a different situation more effectively without worrying about loosing ammunition. Lets assume in a defensive position with access to lots of ammunition if needed.
Stopping power is used to describe the ability of a particular weapon to stop the actions of an individual by means of penetrating ballistic injury. Stopping is caused not by the force of the bullet, but by the damaging effects of the bullet blood loss, etc. The more immediate effects can result when a bullet strikes a critical organ such as the heart or the spine or brain. (or virtually dead in some 7.56mm round cases).
On the Army side of the house, 5.56mm allows for lighter loads and greater quantitity to be carried by an average infantryman. The 5.56 round is not as dense as the 7.62 but is more than enough to maim and cause injury to the target in case its not killed, an injured colleague poses more logistical challenges than a dead one. However as newer weapons of the 5.56 caliber are developed, the velocity of the round will further increase the impact of the round. (From PakDef)
From my knowledge, in a defensive position it's better to have the 5.56mm cause you can keep the enemy combatant at bay before, the air strikes or reinforcements can be called it. Also, know that use of the smaller round it's also vital to hit places of bodily function to render the enemy useless or he will fight back (but it also depends on the range the close quarter combat is taking place). This can be beneficial to capture the combatant to get more intelligence out of him.
As for the 7.56mm:
It can be effectively kill an enemy combatant, not many people live to tell the tale after being hit by it. A short story I can give is about my friend who was in a defensive position (Private. Christoper Scuba), in an alley in Iraq:
While he was bent down and on his knees' next to a Humvee wearing a bullet proof vest, he was struck next on his kidney, the impact of the round actually penetrated the vest making an (1) inch of a hole on his body, causing his kidney to malfunction that he has to get replaced (or) removed, but benefit of this round is it's awesome fire-power, which can cause alot of damage and basically render the enemy obsolete rather than just causing minimal damage to him. With this round some technicals can be taken out which are lightly armored to not being armored at all. (Whats even more deadly is the armour piercing version of the 7.56mm round)