Hunt tests are non-competitive events where a retriever's working abilities as a hunting dog are tested.
Field trials are competitive events where a retriever's working abilities as a hunting dog are tested.
The AKC is by far the largest sanctioning organization of field trials. There is no comparison.
There are a handful of sanctioning organizations for hunt tests. The largest is the AKC. Second is the UKC. The retriever organization affiliated with the UKC is called the Hunting Retriever Club (HRC). That is why you will commonly see it represented online as "HRC/UKC."