Per a request by @MightyPaix here is the commentary I posted during the Sunday lifestream, providing some background on the AK-47 and AR-15. This post is rearranged slightly for easier reading, and I added some more relevant info.

1) What is an AK-47?
The original AK-47 was developed by Mikhail Kalashnikov (who was a tank commander during the war) and was called the Avtomat Kalashnikova 1947 (so automatic Kalashnikov 1947). At that time new weapons were often designated by the year of approval and this is the reason for the "47." The design was updated soon after, and most rifles built in the Soviet Union were actually the updated AKM. Funny enough, they didn't build very many of the actual AK-47.

The design was influenced by the German StG 44 and both rifles appear similar enough that StG 44s are often mistaken for AK-47s in historical photos. Tip: if the photo is of Russians during WWII, they're carrying captured German rifles. Also, Brad Pitt carries one in the movie Fury.

2) Design and patents
Just like with the famous AR-15, the patents on the operating mechanism expired forever ago therefore lots of companies make their own versions of both these rifles, including in the US. They are often referred to as "AK" (instead of "AK-47" ) because these modern rifles use the same essential mechanism but aren't historical reproductions of the actual rifle. In the industry this is called a "pattern" so they are "AK pattern rifles."

AK pattern rifles are popular with some gun owners due to the increased reliability of the gas-piston system over the direct-impingement system used in the AR pattern.

3) Ammunition
The rifle was designed to fire 7.62x39mm ammunition (basically a low powered .30 caliber) and that's still the most common in civilian use. The Russian military swtiched to a new cartridge (5.45x39mm) and a new version of the rifle, designated the AK-74. Current AK rifles are available for the 7.62x39mm cartridge, the US standard 5.56x45mm NATO (like the AR-15), as well as other cartridges.

The 7.62x39mm round is very tapered, so ejection is much more reliable than the AR. The extreme curve of the magazine is created by the large taper of the cartridges. Ammunition is cheap and you can even buy steel-cased ammo from Russia (or, you could before the sanctions).

The 7.62x39mm is also more powerful than the 5.56x45mm NATO round of the AR. For reference, here are some general approximations:
.30-06 (US M1 Garand rifle, WWII) - 3,900 joules of kinetic energy
7.62x51mm NATO (US M14 rifle) - 3,400 joules
7.62x39mm (AK-47, AKM) - 2,200 joules
5.56x45mm NATO (US M16, AR-15) - 1,800 joules
These numbers vary widely based on barrel length, cartridge loading, etc. But you get the point - the scary AR is firing a cartridge less than half as powerful as the standard for WWII.

4) Legality
The legality of both types are the same, as they are currently sold in the US market. Just like the AR rifles, civilian versions of the AK are semi-auto only unless you have a special license.

"Assault weapon" bans include all of the AK pattern rifles in their list of guns to ban, and legally they are treated the same as an AR-15. From a practical legal standpoint, the only difference between the two is the standard AK round isn't commonly manufactured in the US so it's easier to cut off the ammo supply to the public by banning ammunition imports.

Keep in mind there are other rifle design patterns so not all rifles that fire these cartridges are AK or AR type rifles. For example, H&K makes a popular line of rifles that use the same magazines as the AR but are not AR rifles - although they have the same mounting rails so they can carry the same accessories. Every scary black rifle isn't an AR.

If you want to learn more, pick up a copy of The Gun by C.J. Chivers, which is about the development of the AK-47 but also discusses the development of the M16 (the military designation for the full-auto version of the AR-15).

(Note for future arguments: firearm ballistics is one of the most contentious subjects on earth, made worse by the fact that it's not really legal to test guns by shooting people. Take everything said by everyone, including me, with a grain of salt).