I grew up in Seoul and I have experienced all seasons in Seoul more than I could count.
Some places are good to visit any time of the year (like San Francisco, I guess?), but when it comes to Seoul, weather could be a problem. Korea has 4 very distinct seasons. Winter can be extremely cold like -20°C and summer can be extremely hot like 35°C. Typical holiday months in most countries are July and August, but I think you should avoid both months if you plan to visit to Seoul and Korea. It is hot and humid! What I cannnot stand is not the high temperature, but the humidity. Likewise, Christmas season and January and February are not good time to come to Korea either. It is cold, very cold, and cooooooooold!
My favorite months of the year in Korea are September, October, May and June and I think these months also are the best time to travel around Seoul and Korea.
September – October
The best time to visit. Still warm enough to stroll around in shorts and T-shirt (in good weather) and you can enjoy beautiful autumn colors as well. If you love hiking, early November could be good since leaves reach their peak color in early November, but I still prefer October, because you have both relatively warm weather and fall foliage.
May – June
May is festive time in Korea. The most popular one is the Lotus Lantern Festival which is held to celebrate the birth of Buddha. The main reason why I love these months is the scent of the nights. Well, I love strolling through the city in the night in May and June. It is just so lovely to smell the scent of flowers, feel the warm breeze, and maybe sipping a glass of wine or beer while walking in the park or beside the Han river (drinking in public spaces is allowed in Korea).
Well, of course, you could enjoy Seoul any time of the year. Just if you have more flexible options, I think it would be nicer to be in Seoul in these 4 months. Especially, right now in October. I hope everyone in Seoul now bask in the glorious days of the season. 😉