Phoenix Zoo has a fantastic walk-through Squirrel Monkey exhibit that is arguably one of the best primate exhibits to be found in any American zoo simply because it is such a fun experience strolling through a large group of active monkeys. There were 16 monkeys in that enclosure when I was last there in 2011. Bornean Orangutans have a fairly new yet merely adequate exhibit, while Hamadryas Babooons and Mandrills can be found in old grottoes that are once again perfectly adequate but nothing spectacular. Black-and-White Ruffed Lemurs, Siamangs and Ring-tailed Lemurs are all to be found on islands, and there are other primates in the South American section such as Black Howler Monkeys, White-faced Saki Monkeys and Golden Lion Tamarins.
Los Angeles Zoo has a great Chimpanzee exhibit, a decent Bornean Orangutan mesh enclosure and an adequate Lowland Gorilla exhibit that in truth has not aged very well. According to their website there are Siamangs, Buff-cheeked Gibbons (both in those atrocious roundhouses with very little vertical space), Mandrills, Ring-tailed Lemurs, Black Howler Monkeys, Kikuyu Colobus, Geoffrey's Black-handed Spider Monkeys, Squirrel Monkeys, Francois Langurs (in a great exhibit) and of course the legendary Red Uakaris. I'm not sure how accurate the website is as I can remember Coquerel's Sifakas as well, but most zoo websites are poor when it comes to updating animal lists.
A tough call!!! The list of great apes seems to swing things in favour of Los Angeles, but while L.A. has top-notch exhibits for chimps and langurs, most of the rest of the enclosures are a tad disappointing. In fact, it is a huge black mark against the zoo for having the two gibbon species kept in 1960s-era cages. Phoenix's walk-through Squirrel Monkey exhibit is so impressive that even with less primates overall I think that I'd vote for Phoenix. At least 10 primate species in mainly good but unspectacular exhibits, and the Squirrel Monkeys represent the cherry on top of the cake.
I visited Phoenix Zoo in 2008 and 2011; I visited Los Angeles Zoo in 2008 and 2017.