There are two things I noticed from our (Indian military's) perspective:
The 5.56 bullets are better in close to medium range combat. It can kill an enemy or terrorist at a range upto 300-400 meters, if we assume all the ideal conditions around (winds, orientation etc which impact a bullet). Equipped with softer heads, some of the latest ones of these types are designed to 'flat out' inside the enemy's body like a cake being splattered face-first. Once the end is flattened, the trajectory of the bullet changes and goes like an upward arc within an enemy before exiting, causing severe internal damage and trauma. Its main drawback is its range which makes it pretty ineffective when fought in Himalayan terrain due to the obstacles and the LOS conditions in combat. This forces the soldiers or commandos (using non-INSAS weapons) to focus their attention on targeting the right kill spots: throat, heart and head/face.
Now in the case of 7.62 bullet, it is heavier which carries more momentum when fired and therefore can go much further than their smaller NATO counterparts. Its size and force causes devastating trauma, ensuring an instant kill(Except for a Russian soldier who was once fired in the head by an AK bullet, but it somehow got stuck in his forehead skin rather than cracking his skull open. He was smiling while his comrade pulled the bullet out with pliers.
).
Most CT units prefer AK series as the ideal weapon especially in our tense areas where hit-to-kill is the policy. Here the main problem is carrying those rounds. They're heavy and are really difficult to manage in automatic fire.
One of my elder cousins is in RR unit and though they're trained to use AKs like toys, he does admit that its difficult to control it in full automatic fire. Especially in an ambush firefight where suddenly you are attacked from somewhere and you don't have the time to control your weapon, and aim for the attacker in a fraction of a second.