I've never connected through JFK, but I did fly in and out of it a few months after flying out of ICN.
I would say that ICN looks spotlessly clean and modern, with excellent customer service, while JFK looks rundown and drab, with surly or indifferent customer service (even though I lived on the East Coast in a previous life and know that New Yorkers are not really rude, just upfront about whatever emotion they happen to be feeling at the time, which can be good or bad, their customer service leaves a lot to be desired.)
However, my least favorite airport so far is LAX. Unfortunately, it's often the best choice price-wise, since competition among the half-dozen Asian airlines that serve the immigrant populations makes for low airfares to Asian destinations.