How does the Army differ in the States as opposed to being stationed in Korea?

Several things that won't change anywhere in the Army, Your General Orders, P.T., S.Q.T, P.M.C.S. and all the other Army letter words with no vowels. ;-)
What will change is the people you work with, the people and culture surrounding the base, and depending on the location, the weather. What's important not to change... You being the best soldier that you can be and taking charge of your career no matter where you are assigned.

If you do well in Korea, you'll be fine in CONUS. USFK/8A policies are much more strict than at most CONUS bases, although those USFK/8A policies have loosened over the years with new command teams. I loved my time in Korea as well, so I extended a few times and finally recently PCS'd after 4-years. One thing I miss very much from Korea is the highly efficient public transit system and the safety. Within a few months of being back at Fort Bragg, my wife and I witnessed a violent crime committed in broad daylight in Fayetteville... In Korea, you can walk down just about any dark alley at any time of the night, and never have any issue from the Korean Nationals.

If you like it in Korea, and you aren't too close to your DEROS date, then I would recommend seeking extension through the Assignment Incentive Program or Foreign Service Tour Extension. Your continued service with the Red Dragons will be very helpful to the mission, due to the yearly loss of personnel creating vast gaps in experience. While you are there, make sure you catch a Korean baseball game, even if you don't like baseball. Game tickets are cheap and the crowd experience is more entertaining than the field play.

The Army as a whole is what you make of it, regardless of where you are. Every base is different depending on what formation type is there, local laws, who actually owns the land the base occupies, and how large of base it is. I prefer smaller bases because there tends to be less stupid rules.
Bragg, lot of stupid rules, but it's fairly easy to get good training. Good luck and remember there is no running on Ardennes.