I've been to both - and there is a difference -
Gogi - Korean BBQ
Gyu-Kaku - Japanese BBQ
While the concept of a BBQ pit is the same - the flavors are different.
Gogi - the staff actually does most of the work with the grills and you get TONS of sides (bapsang). It's not all you can eat, but the price is very reasonable - about $40 a head (there was 6 of us), and we got 3 different meats, a couple of appetizers, a couple of pitchers of beer.
Gyu-Kaku - I don't think we got any side dishes, we picked the meal deal which came with fried rice, a couple of appetizers - but there was a larger variety of meat. The staff was pretty much there to drop off the meat - we did most of the grilling. We only got 1 pitcher of beer, and I believe the bill came out to $80 a head (6 ppl).
From my understanding with the Gyu-Kaku and the all you can eat meal - there's a time limit of 90 minutes ... you may want to check on this.
You may want to take in consideration of the ages of the kids, and if they're going to need to babysit little curious hands or not.