I've read both the AZA and ZAA standards cover-to-cover. The AZA standards are deliberately vague on a lot of things, the rules basically being "your peer zoos will determine whether you meet this standard." The ZAA is clear and explicit, but has basically a few levels of care depending on rough categories of animals. The AZA also covers a lot of things beyond animal care, mostly related to paperwork and having enough employees. I scoffed at those until seeing the Capital of Texas Zoo, and I'm still skeptical of the rule that says for all of your animals for which the AZA has a breeding program, you surrender decisions about the breeding of those animals to the coordinator for those animals.
I've been to only one ZAA zoo and it's much better than any unaccredited zoo I've been to in terms of animal care. Getting some form of accreditation seems to guarantee minimal standards; the Austin Zoo which is accredited by a sanctuary organization is also better than unaccredited zoos. The best zoos in the US today are all AZA zoos (though as far as I know Parrot Jungle was never AZA, but until the new owners decided to de-emphasize parrots it set the standard for parrot care and breeding in the US), but I think that that may be in part explained by the fact that for zoos with a lot of resources, AZA gives access to animals that would otherwise be hard to come by.
A lot of people who run smaller zoos got their start as exotic animal hobbyists, and the AZA has kind of a double standard where private breeders are concerned. It's OK (against policy but fine in practice) for AZA zoos to get animals from private breeders, even really shady ones, but it's better to euthanize surplus animals than sell them to private breeders. (I've seen the AZA described as a cartel, and I think that in some ways it kind of is, though it's a relatively benign one as cartels do.) I know of at least one person who runs a ZAA zoo (which I haven't been to) but refuses to seek AZA accreditation because it would be validating their policy on private breeders.
I think that in general the AZA standards are better than the ZAA standards but the ZAA standards read as if written by a committee of bankers and the AZA standards as if by a committee of bureaucrats. If I somehow managed to start a private zoo, AZA accreditation would be an ultimate goal which I'd delay as long as possible while seeking ZAA accreditation immediately. AZA accreditation seems like would add a lot of expenses to a small operation (in fact I think that that may be part of the idea) and it would also require surrendering a fair deal of control over my hypothetical collection, but it would also mean access to the AZA zoos' surplus animals.
