The Kalashnikov is a masterful design. It is simple, easy and cheap to manufacture by any of a number of processes (stamping, milling, forging), and literally can be buried in mud for days and remain operational. Sure, it's not as accurate as the AR platform (e.g., M-4, M-16), but what good does that do you when your gun doesn't work?
In fairness to the AR, it is a highly advanced, very reliable weapon when properly maintained. Its early reputation for failures can really be blamed on the Army's failure to properly train the cleaning processes or issue appropriate cleaning tools. With good cleaning, it's a marvellous design. The Kalashnikov, on the other hand, can be thrown at a raw conscript and will remain operational basically no matter what.
Interestingly, this dichotomy goes back much further. Prior to the AK there was a short run of the SKS (mostly adopted by China later, and also designed by Kalashnikov), and before that the (in)famous workhorse of both the Czar and the Glorious Revolution, the Mosin-Nagant. Compare this to the weapons used by the US over this period, viz., the Krag, Springfield, and M1 Garand---especially the latter!
Tangentially, it's quite impressive to consider that, in an era of bolt action infantry rifles (the Brits, Germans, French, Russians, and Italians had the Enfield, K98K, MAS-36, Mosin-Nagant, and Carcano, respectively), the US led the way towards modern rifles with the Garand. Helluva gun, and fun as hell to shoot.