Having visited four of these facilities (all but Tanganyika and Tennessee Safari) I would rank as followed:
1. Southwick
2. Monkey Jungle
3. Richmond
4. Zooworld
I rank Southwick first because of its collection and enclosures. If this same thread was around a few years back Southwick would not be #1. The zoo has drastically revamped its facilities adding beautiful exhibits for its big cats, small primates, elephants (visiting) and especially the chimps. They have done away with most of the balen cages and have combined several yards to make one large habitat for a single species not small yards with numerous species. The collection is also impressive with over a dozen primate species (ranging from marmosets to chimps), a decent reptile collection, a large parrot collection and numerous hoofstock species. Hoofstock highlights include white rhino, bearded pig and giraffes. Also, the owner of the zoo is usually either in the giftshop or the education building and loves to talk animals and the zoo.
Monkey jungle is really cool to visit wether you want to walk through the free ranging monkey exhibit (Howlers, Capuchin and Squirrels) or watch the macaques dive into the water for treats. The reason I put it second: its collection is primates and birds. Not the most diverse collection, but species like wooly monkey are always a treat. Also it always kind of bothered me they had a lone gorilla. Its a decent enclosure but its kind of weird seeing a single silverback.
Richmond barely squeezes ahead of Zooworld because of its collection. It is a very impressive collection, mostly mammals, with highlights including white rhino, mishmi takin, brazilian tapir, snow leopard, cheetah, asiatic black bear, orangutans, chimps, Allens swamp monkey, African penguins and galapagos tortoise. They have a mix species with giraffe, bongo, greater kudu, thompsons gazelle, white stork and crown crane that was quite impressive. The main issue: enclosures. Some of the worst, barren and sometimes unsafe enclosures I have seen. Most enclosures were chain link which made it difficult to see them. Some of the hoofstock yard were barren and dusty while others had debris and stagnant water. The tapir enclosure I could lean into which I am sure people did because of a calf born around the time of my visit and you could easily pet the RRH's and penguins if you so desired. I didn't really see any enrichment either. Also, and this may be my biggest pet peeve with some zoos, no signs. I looked into primate enclosures one after the other that had nothing saying what was on exhibit. The were Angolan colobus, Allens swamp monkey and Reed's titi and I know that because I have seen them before. Your average visitor would have no clue.
Zooworld is interesting because it has many characteristics of a road side zoo except the collection. The zoo enclosures are made up of balen cages, chicken wire fencing and concrete. Very typical of a subpar zoo. The collection however is interesting. Up until recently the zoo had a Sumatran orang who lived with an orange tabby cat. The orang passed earlier this year. Next to her, where siamangs, ringtail and black lemur. Other primates included mandrill, diana monkey, red handed, golden headed, golden lion, geoffferies, cottontop tamarins, and callimico. Cats include jaguar, puma, serval, lions, four color phases of tiger and Sumatran tigers. Yes, Sumatran tigers. I confirmed this with the tiger SSp a few years back. They also have spectacled bears, a giraffe you can feed and a timber wolf you can pet during certain shows.
Other non-accredited places I would add:
Six flags: Great adventure- Safari park with large herds of white rhino and African elephant, lots of African hoofstock including white tail wildebeeste, Syrian brown bears, Amur tigers and more.
Reptile Gardens- One of the most complete collections of reptiles in the country.
Alligator Adventure- Once again, great herp collection with Tomistoma and Black mamba and one of largest crocs in the country, Utan. Also birds including cassowary and mammals (tigers, small cats, otters, zebra and more).
Catoctin Zoo- Located in Maryland, the collection includes sun bears, big cats, primates and a good herp collection. They just added a safari area to the zoo.