Kobe, in short, is a variety of Wagyu. Wagyu, loosely translated, means “Japanese cattle” (“Wa-” meaning Japanese or Japanese-style, and “-gyu” meaning cow or cattle). So “Wagyu” refers to any cattle that is bred in Japan or the Japanese-style. Kobe beef is comprised of a very particular strain of Wagyu called Tajima-Gyu that is raised to strict standards in the prefecture of Hyogo. (Hyogo’s capital city is Kobe, thus the name).
Every ribeye is a steak, but not all steaks are ribeyes. A similar rule applies to Kobe and Wagyu beef: Every Kobe steak is Wagyu, but not all Wagyu beef is Kobe.