I'd pick Korean.
United in-flight service varies from indifferent to aggressive.
United food in economy is sub-par. Though after a number of UA flights some years ago I worked out that if you pre-order a Hindu Veg special meal you get a pretty decent plate of curry and of course get your food before everyone else.
Korean will fly you in more comfort: their A330 and A380 aircraft are marginally more comfortable in economy class than the narrower, densely configured 787. For the return flight on the 777, Korean is one of the handful of airlines that has retained the more civilised 9 across seating on that aircraft rather than 10 across. What that means is the same number of seats in each row on a Korean 777 as UA cram into the 787 which is 550cm in diameter vs 590cm for the 777.
The stop in Seoul might be good to let your kids run around and let off some steam mid-trip. The transit process is very easy and Korean Airlines just moved into the brand new terminal 2 at Seoul airport.
The food on Korean is great - if you like Korean food. Western options are edible but unremarkable.
One area that Korean falls short of UA is the choice of English language in flight entertainment. There's a bit there, but not a lot. And another shortcoming is the weird service timings on trans-Pacific flights where they wake the whole cabin up for a meal service about 4 hours before landing, clear it away and switch the lights off again. Oh and they insist on blinds being closed on day flights; too bad if you like to look at the planet from above.
Despite those shortcomings on Korean, I'd still pick them over United.