Best part of Korean BBQ that Yakiniku doesn't have: wrapping your BBQ meats in bite sized lettuce bundles (ssam), with kimchi, garlic, rice, etc.
It's claimed the present style of Yakiniku is derived from Korean BBQ restaurants that opened in Osaka and Tokyo around 1945, accommodating Korean style BBQ to Japanese tastes.
http://en.wikipedia.org/wiki/Yakiniku
The difference between Korean BBQ and Yakiniku in Vancouver:
Korean BBQ will commonly have lettuce (ssam) to wrap your meat, and side dishes (banchan) such as kimchi, sweet potato, bean sprouts, etc. The dipping sauce will likely be ssamjang (chili pepper and fermented soybean), and meat could be marinated in typical Korean flavours like that of sweet bulgogi, or spicy gochulchang (chili pepper).
In Vancouver, Yakiniku will seldom come with included side dishes (but sides can be ordered), and the meat is meant to be eaten on its own. Included dipping sauces are usually oil with salt and pepper, a soy based one, or a miso type. Marinades could be a typical Korean style sweet bulgogi, or miso, and more commonly unmarinated in Yakiniku than Korean BBQ.
The lines are blurry in Vancouver as many Yakiniku here are Korean owned, and they include some Korean items on their menu. Conversely, many Korean BBQ restaurants here have Japanese items on their menu due to popularity.