On my first visit to Korea over Christmas in 2004 we did two excursions to sites outside Seoul. We went to Gyeongju for Korean history. And we went to Seoraksan for Korean nature.
Seoraksan is the most popular National Park in Korea. It is situated on the Eastern coast, roughly a 3h drive by bus from Seoul (depending on the traffic in Seoul). Coming from modern Seoul it's quite a contrast to see the rural side of Korea and the mountains from the bus window as you venture to Sokcho, the gateway to Seoraksan.
The National Park itself is surprisingly close to the coast. Essentially, it starts directly on the city limits of Sokcho, maybe a kilometer off the coast, and then climbs fast.
In the park, the most known sites are a giant buddha at the entry. And the unmovable stone. We did some hiking and site seeing. I wish we had had more time to venture deeper into the park as the nature was great, even in winter.
OUV
Having been to Korea twice, I feel that the country has two defining features:
The coast
The mountains
Both are not really represented on the list and in my eyes deserve to. For the coast, the tidal flats are nominated. For the mountains, I would be perfectly fine with adding Seoraksan. To me, this is one of the great national parks of the world and deserves inscription. I could even see the case for a mixed inscription as the park is dotted with buddhist temples and shrines.
For Koreans the peaks in Northern Korea are the even greater ones, specifically Mount Kumgang on the other side of the border. It may be the hard to get to factor that adds to the longing. So as mentioned by Kyle, a transborder nomination of Mount Kumgang and Mount Seoraksan would have charms. Or the whole mountain range stretching the Korean peninsula.
Getting There
We took a bus from Seoul and stayed for one night in Sokcho. From Sokcho, you can take a shuttle bus to the park entry.
While You Are There
Depending on the state of affairs between North and South Korea it has been possible in the past to do trips into North Korea. You could visit specifically Mount Kumgang with eased entry requirements. At the time of writing (2019), this seems to be closed down. Sokho itself is a coastal town with a beach, I seem to recall. This wasn't really an option in winter.
Notes
There are different romanizations of Korean. 설악산 to my knowledge in the site's name in current romanization should be Seoraksan.