The biggest decision you are going to have to make when moving over to Korea is whether or not you want to teach in a private school(Hagwon) or public school. Trust me the difference is HUGE.
Job - If you go to a public school you will work with the Board of Education as an ALT, or, alternative language teacher. Because laws in both Korea and Japan wont let foreigners become actual teachers. So to get around this, there will be a Korean teacher in the class room and you will simply be assisting him. Most of the time you will actually be the one teaching the class, having a Korean in the room is to dodge a loop hole in the system to allow foreigners to teach. You will also most likely teach at multiple schools in the area, not just one. However, at a Hagwon (pronounced hogwan) you will be the homeroom teacher. You will be the only one in the class room teaching, you will make the lesson plans and you will have a class all day everyday of the week as opposed to seeing a class once a week with multiple classes all week.
Duties - Another difference would be class size, at a Hagwon, you will most likely have 10-15 students. At a public school there can be up to 40 per class. As mentioned before with public schools you aren't directly responsible for your students, you don't really have to monitor their progress and since you will be teaching at multiple schools and many classes with 40 students this is a good thing. However at a private school you will be expected to do a whole lot more. You must make the lesson plans every hour of every day, field trips, grade cards and progress reports, you will work more hours, most likely teach 3 classes with 15 students each day so thats roughly 45 students you are in charge of and responsible for. In my experience I would say Hagwons are more involving and fun, public school teaching can get very dry and boring fast as you be teaching the same thing multiple times a day, and then in a few months when the next trimester starts, you will restart your material all over again.
Pay - Hagwons generally pay more but its only a fraction, not something worth taking into consideration. Plus the more time spent teaching the more money you earn each year. Both private and public school teachers can get well over $3,000 a month in a few years of work.
Employer - Your employer will be the BOE if teaching in public schools, this is a very very good thing, as the law and government watch over the BOE very closely. However if you choose to teach at a Hagwon, be warned, no one, and I mean no one, monitors there actions. Working for a Hagwon is a complete and total gamble, I've heard stories of people working at good Hagwons, but that was online and I doubt they are real. I have never met anyone who enjoyed their time at a Hagwon, they do very dirty business. But thats for another blog. Something to keep in mind is a private school is allowed to make their own contract and this contract is allowed to say just about anything. So Make sure to READ your contract carefully.
There is plenty more that could be said, but I rather this not turn into a huge blog post, so there may be more to come. I hope this helps in making your decision of where you might want to teach in South Korea.