There has long been debate over which caliber is superior between 5.56 and 7.62. Each has it's strengths and weaknesses. In the end, both will get the job done. Here we can give you a little history, some facts, and hopefully you continue on with a bit more knowledge for your brain housing.
The .223 Remington was the inspiration for 5.56 ammo. The bore of the 5.56 measures .22 . 7.62x39 ammo was created from the .308 Winchester cartridge. It was designed to replace the U.S. Military's .306 Springfield rifle cartridge. Its bore is .30 which is considered a large caliber.
The main draw for the military to the 5.56x45 caliber is that troops could carry a substantially larger amount of rounds on their person than the 7.62x39mm due to 5.56 having a smaller / lighter footprint.
The larger 7.62 round is generally the more powerful option. Since it has more stopping power than the smaller 5.56x45, the 7.62x39 cartridge is the preferred caliber of military sniper teams. The heavier the bullet, the less that crosswinds will have an effect on the trajectory at longer distances. On the other hand, the lighter 5.56 travel faster w/ a flatter trajectory. The U.S. military has been using 5.56 for their most utilized firearms since the Vietnam War.
The 5.56 NATO round is most commonly associated with the AR15, whereas 7.62 is primarily used for the AK47 platform of weapons. Since the AR15 is the U.S. favorite, ammo for it is more readily available and cheaper than the larger 7.62. There are some AR15's from manufacturers like Radical Firearms, that can fire 7.62. The same goes for AK's, some can fire the 5.56 caliber, so you're not necessarily locked into one or the other if you like the aesthetic of an AR over an AK or vice versa.
In the end, you can't really go wrong either way. 5.56 might be the way to go if you're a smaller shooter and can't handle as much recoil, or you might want to really FEEL the power of a 7.62. As always, it pretty much comes up to your preference. Stay safe and have fun!