Logistically I think the only difference between going to the Hyatt Regency and heading into town would be the length of time on the bus (assuming you take either a limo bus or a shuttle bus to your hotel). But coming from Sydney I will assume your body clock is 2 hours ahead of Korea time. So let's say you make it to your hotel by 9pm local. It will feel like 11pm? But the beauty of this will be in the next morning where I assume you might get an early start in the morning.

There might not be enough time to go visit a palace or somewhere cultural the next day, but just being with the kids and enjoying breakfast together in a foreign country might be fun. Maybe take a selfie picture then? I don't recall who said or wrote this but sometimes it is not the destination but the journey that makes it memorable.