Changdeokgung Palace is a UNESCO World Heritage site. Thus, since we had time, we decided to take a look. It's actually a lot smaller than Gyongbukgong, so if you have time for only one, see the former. The architecture seems similar, though it's a bit more free-form rather than being on a strict grid.
There is a large garden in the back but it requires a separate admission. I heard that it's very nice. But we didn't go since we visited during the winter!