Omaha doesn't have a dedicated South American section, but it does have a lot of species from the realm - mostly living in Lied Jungle and Kingdoms of the Night (and Simmons Aviary for some of the birds). Mammal species of note are: Baird's tapir, 5 species of bats, owl monkey, red-backed bearded saki, 3 species of armadillo, and lowland paca. Its herp collection is pretty good too, with 2 species of basilisk, yellow anaconda, Haitian boa, and American crocodile. The Neotropical bird list is somewhat less impressive (especially compared to some other American zoos), none of which are noteworthy except gray-necked wood rail. Collared Peccary Baird's Tapir Ocelot Coati Seba's Short-tailed Fruit Bat Greater Spear-nosed Bat Greater Bulldog Bat Common Vampire Bat Pallas' Long-tongued Bat Douroucouli Black-headed Spider Monkey Black Howler Monkey Red-backed Bearded Saki Common Squirrel Monkey Hoffman's Two-toed Sloth Nine-banded Armadillo Three-banded Armadillo Screaming Hairy Armadillo St. Vincent Agouti Prehensile-tailed Porcupine Nutria Lowland Paca Burrowing Owl Turkey Vulture White-faced Whistling Duck Northern Pintail Chiloe Wigeon Cinnamon Teal Bahama Pintail Blue-winged Teal Northern Shoveler Ring-necked Duck Cattle Egret Inca Tern American Flamingo Roseate Spoonbill Scarlet Ibis Gray-necked Wood Rail Crested Screamer Scarlet Macaw Blue-and-Yellow Macaw Blue-grey Tanager Silvery-beaked Tanager Toco Toucan Mexican Spiny-tailed Iguana Green Iguana Argentine Black & White Tegu Collared Tree Lizard Green Crested Basilisk Plumed Basilisk Cantil (species unknown, probably A. bilineatus or A. taylori) Central American Tiger Rat Snake Emerald Tree Boa Amazon Tree Boa Jamaican Boa Haitian Boa Yellow Anaconda Dwarf Caiman Spectacled Caiman American Crocodile Mata Mata Amazon Milky Tree Frog Mexican Leaf Frog Smooth-sided Toad Yellow Backed Dart Frog Panamanian Golden Frog Green & Black Dart Frog Hourglass Frog Smokey Jungle Frog Exhibit wise, the primates and tapir live in enclosures like these in Lied Jungle: Spider monkeys (and maybe also capuchins? Can't remember) also live on an island outside in the summer: A few species (burrowing owl, turkey vulture, coati, ocelot, peccary) live in Desert Dome, which is heavy on North American animals that originate from or overlap with the Neotropical realm. The peccary enclosure is well-landscaped but very small, and the ocelot enclosure is essentially a long narrow cave with a mesh barrier that situated along the wall behind it. I get that it's probably supposed to be immersive and space-efficient, but I don't see how one could get even a half-decent view at a cat in a dark cave from across the peccaries.
And here is Berlin Tierpark's list, according to ZTL. Both zoos are about evenly split on mammal species (Omaha 22 to Tierpark 21) and reptile species (17 at both), but Omaha wins handily on amphibians (Omaha 8 to Tierpark 3) while Tierpark reigns supreme on birds (Tierpark 50 to Omaha 22). Some notable species from Berlin Tierpark are: margay, yellow-breasted capuchin, bush dog, Venezuelan red howler, vicuna, 2 caracaras, several tanagers, many amazons and macaws, Chilean blue eagle, harpy eagle, and 3 vultures; its herp collection is more generic. Azara's agouti White-faced saki Spectacled Bear (Andean Bear) Chacoan peccary (Tagua) Degu (Common Degu) Yellow-breasted capuchin (Buffy-headed tufted capuchin) (Weeping capuchin) Patagonian mara (Patagonian cavy) (Mara) Guanaco East Bolivian Highland yellow-toothed cavy (Muenster yellow-toothed cavy) Chacoan mara Central American margay Venezuelan red howler Vicuna Bush dog (Savannah dog) Unstriped tamandua (Yellow tamandua) (Long-tailed tamandua) Maned wolf Silvery marmoset Southern three-banded armadillo Lowland paca Golden-handed tamarin (Midas tamarin) (Lacepede's tamarin) Northern Raccoon (Common Raccoon) Laughing gull Great green macaw (Buffon's macaw) Mountain caracara Black-chested buzzard-eagle (Grey buzzard-eagle) (Chilean blue eagle) Blue-throated macaw (Wagler's macaw) Blue-headed macaw (Coulon's macaw) Northern festive Amazon (BodinĀ“s amazon) Rusty-barred owl Chaco owl Chilean flamingo Lesser rhea (Darwin`s rhea) Green-mantled tanager Striated caracara (Forster's caracara) Yellow-faced amazon (Yellow-faced parrot) Blue-and-yellow macaw (Blue-and-gold macaw) Golden-headed manakin Blue-cheeked amazon (Blue-cheeked parrot) Grey-winged trumpeter (Common trumpeter) (No Subspecific status) Grey gull Green-winged macaw (Red-and-green macaw) Southern screamer (Crested screamer) Harpy Eagle Hyacinth macaw Yellow-billed amazon (Jamaican yellow-billed amazon) Jandaya parakeet (Yellow-headed conure) (Jendaya conure) Common burrowing parrot (Lesser patagonian conure) Lesser yellow-headed vulture (Savanna yellow-headed vulture) King vulture Crested oropendola American flamingo (Caribbean flamingo) (Red flamingo) Western cattle egret (Buff-backed heron) Peregrine falcon (No Subspecific status) Red-legged honeycreeper Northern shoveler Southern mealy amazon (Nominate subspecies) American white pelican Turkey vulture (No Subspecific status) Spangled cotinga Black-necked swan Glossy ibis Sunbittern (No Subspecific status) Northern pintail Rosy-billed pochard (Rosybill) Greater Antillean bullfinch Bay-headed tanager (Green tanager) (No Subspecific status) Red-crested finch Red-fronted macaw Scarlet macaw (Red-and-yellow macaw) Eastern brown pelican Paradise tanager (ssp. paradisea) Golden tegu lizard (Black tegu) Blue spiny lizard Colombian rainbow boa Mata-Mata (Matamata Turtle) Garden tree boa (Amazon tree boa) Common spiny-tailed iguana (Black iguana) Common green iguana (No Subspecific status) Honduran milksnake Common northern boa Red-footed tortoise Cuban slider (North Antillean slider) (No Subspecific status) Yellow-footed tortoise Uracoan rattlesnake South American rattlesnake Red-cheeked mud turtle Northwest Neotropical rattlesnake Pueblan milk snake Axolotl Mission golden-eyed tree frog (Amazon milk frog) Sabana surinam toad I don't know much about the zoo's exhibits or layout, so here's just a sampling of relevant photos for mammals and birds: New World Vultures Harpy Eagle Macaws Flamingos (with camels in the background?) Howler/Capuchin/Agouti Outdoors and Indoors Spectacled Bear Margay (featured alongside non-relevant exhibits) Chacoan Peccary