I read a few articles on weapon magazines in 80s about this.

1) while 5.56 is lighter, when it impacts with the human body the bullet will often impart the so-called "tumbling effect" (i.e. instead of keeping its straight course it basically tumbles around inside your flesh - and may create secondary shrapnel if it hits a bone, causing a lot of damage and hydroshock). So yes, if you want to hit someone behind light cover, like a wooden door, 7.62 is more effective, but if you hit directly the resulting damage is equivalent.

2) yes, the bullets are lighter/smaller - this means your average infantryman can carry more magazines as part of his load, so you may be able to fight a bit longer than you opponent packing 7.62.

EDIT: added some words to clarify the tumbling effect.