OK, let’s keep it simple. All Kobe is Wagyu, but not all Wagyu is Kobe. Kobe is a registered trademark developed in Japan around 1983 to distinguish the strict rules surrounding the growth, feed, and finishing of the 4 types of Wagyu cattle in Japan. True Kobe is ridiculously high in fat content (look at the picture!)! but oddly, much higher in unsaturated fat producing oleic acid, the stuff that makes for good cholesterol.
Beef labeled “Kobe Style” or “Wagyu” in the US is usually a cross-breed of Wagyu and Angus.