I haven’t been to Korean BBQ in NYC in a long time. My knowledge is mostly from browsing reviews and stuff online. But generally speaking there are 3 kinds of Korean BBQ in Manhattan: Old school, casual new school, and premium new school. Cote as people mentioned above is a premium new school joint offering some really premium cuts. If money is no object, Cote and Don’s Bogam seem to be considered the best in the city.
The “new school” joints change some of the classic formulas of KBBQ and try to do some things others aren’t doing. The biggest difference is that you won’t be getting the massive array of side dishes that’s a trademark of Korean BBQ restaurants. If that sounds like your thing and want something a little more fun and casual, Kang Ho Dong Baekjeong is a really popular spot that’s one of the original “new school” chains from Korea. They have some fun things there like an egg moat and the shaken lunch box thing.
Also, none of these places use charcoal, so if you want that charcoal experience, you’ll only find it at wonjo (as far as I know).
You might also want to consider doing Japanese Korean BBQ, I hear Takashi is really great, but quite expensive. I’ve been meaning to get my ass over there for years.