It depends on your tolerance for cold. Korea does get pretty cold, especially for someone from a warmer climate. Coming from the Northeast US, it hasn't been too terrible for me, but my friends from warmer places do complain.
That being said, it can be bearable. For one thing, you can always do some cheap shopping (as long as you aren't a big person) for jackets, scarves hats etc., though of course I would come prepared.
About the snow: Korea has pretty lame snow for the most part. Korea gets most of it's precipitation during a rainy period of about a month or so in the summer (it's just been starting up). There might be a decent snowfall once a year. Other times, it's just an inch or two or less and gets nasty in the city. I wouldn't count on snow. Though, Gangwon-do supposedly gets nice plentiful snow (quite mountainous there) but I've never been in the winter.
About 4-6 weeks, honestly, I think you may get bored; Seoul and Korea only offer so much. Especially in the Winter, you may get a little wound up. Even with jackets and such, you're not going to want to spend all your time outside And winter is not the best time to go. Autumn is the definitely best time, followed by Spring. However, I also understand that you may have limitations on when you can visit etc. But I would encourage you to re-think spending 4-6 weeks. The other thing is that Korea does not offer a lot of diversity (there's a huge amount of pride in its homogeneity and uniformity). There's a chance you'll find it repetitive.
I'm not trying to scare you off from Korea. I just don't know about 4-6 weeks in the winter.
There are a few nice things about the winter. Hot street food is one of them. I'm not a hug fan of Korean street food, but it's pretty good in the winter. Watch out for roasted sweet potatoes (usually only sold in winter) and ho-ddeok which is a greasy fried yeast dough disc with sweet fillings like brown sugar, cinnamon and nuts. Another favorite is bungeobbang which are hot pastries made in a special pan in the shape of fish and are stuffed with sweet red beans or sometimes custard or even spicy vegetables.
Another nice thing about the winter is a decrease in crowds! Everyone is trying to stay inside, so if you go out and brave the cold you'll have more elbow room.
Good luck to you with whatever you decide what's best for you! Please feel free to contact me with any other questions either before you go or when you're in Korea.
-been living in the Seoul area for 3+ years.
