5.56mm rounds are a lot easier to carry in bulk than 7.62mm.
5.56mm rounds are quite deadly up to 200m, which is generally the farthest distance most firefights take place at. Many soldiers in Vietnam were quite amazed at the little round's lethality.
In the mid 80's 5.56 NATO rounds were updated to promote a "tumbling" effect when striking their target, thereby imparting a LOT more energy into the target.
5.56mm on the whole are probably cheaper than 7.62mm to produce.
AK47 rounds are 7.62X39mm, which isn't a really powerful round for a .30 caliber, compared to the 7.62 mm NATO round, or the earlier .30-06 round of the M1 from WWII.
There is a variant of the AK47, called the AK74 which uses an even smaller 5.45mm bullet. When used in Afghanistan in the 80's some people believed they were using poisoned bullets, they were so lethal.
AK has a deserved reputation for unbelievable ruggedness and dependability. Simple and durable, they embody most of the traits of a first class military rifle, except stunning accuracy. M16 was always more accurate than the AK, but early models in Vietnam were underdeveloped, and many of the rounds used in the early part of the rifle's existence were shipped with the wrong powder, causing excessive cycling rates, overheating and jamming. This is why many soldiers in that conflict often used captured AK's, or tried to hold onto their older M14's, another rifle with a bit of a checkered history. M16 has matured over the years to become one of the better military rifles available. It's no FN-FAL, but it isn't the jammy door-stop it used to be.
I wouldn't want either of them pointed at me in anger.