Lisa had a good note, but I thought I would expand on things. AKC is the much "bigger" organization, and the main one for most breeds. UKC is more prevelant for the rarer breeds, and that's where a lot of them get their start. One example is Pit Bulls, they are not an AKC registered breed, but there are a ton in UKC. Other breeds that are "big" in UKC are Belgian Turvs, Toy Fox Terriers, and American Eskimos, and so on. Basenjis are primarily an AKC breed, there are probably only a few handfuls that ever do anything with UKC.
UKC's goal is to be more of a "family" organization. At their shows they don't allow professional handlers on a dog at any time, unless they own the dog. And usually it's a bit more casual and the shows are much smaller. They really encourage the juniors and most clubs stop all action when the jrs are being judged so the whole family can watch and cheer them on, and folks are not busy in other rings. Also they don't get as crazy with the grooming, they try to keep the dogs in a bit more of a natural look. And they don't allow bait or toys in the ring, unless each specific judge ok's it at the time. Many judges do these days though.
As for the skinny on Basenjis, it's usually a lot easier to finish a dog in UKC. But a big reason for that is because very few of them show UKC, so you don't have a lot of competition. There aren't "majors" and things like that to earn your title. The closest thing to a major is that there must be at least 3 champs to get a leg towards a grand champ. Some people show UKC because their dogs don't do well in AKC. Others show because it's something extra to do after AKC, since many people can't afford to campaign a special. And I think more and more people are going to shift over to UKC because of AKC's increasing fees and lack of real support for responsible breeders.
But I enjoy them and there's lots of nice folks there. The ring procedure is basically the same, the entry fees are cheaper, and there's usually 2 shows each day. So I like to check them out once in a while when there's a close one or others I know are going. But I'll still focus on AKC.