Is there a difference between them? I have looked at some videos of both and the only two differences I can see is that the hunter paces take place in the US and most of the course is on trails through woods, while team chases take place in the UK, mostly in open fields.
I have only been to a few hunter paces, and they were held at eventing barns on the cross country course. They were local activities, organized by the local Pony Club chapter, with other people welcome, and costumes awarded prizes.
There is no local hunt in the area, so no hunt country. I was interested to see that the hunter paces on YouTube passed through so much wooded territory. It looked basically like a very fast trail ride with jumps. Is that typical hunt country in parts of the US?