Texas only regulates the apes.
Harris County regulates nonhuman primates by requiring a certificate of registration and other requirements such as enclosure size, etc. ***Note that this is for the unincorporated areas of the county, so it should not apply if you are within city limits. Might want to double check though.
HARRIS COUNTY’S REGULATIONS RELATING TO THE KEEPING OF WILD ANIMALS IN THE UNINCORPORATED AREA OF HARRIS COUNTY, TEXAS
section 5 REGISTRATION of Dangerous WILD ANIMALS and WILD Animals
A. The owner of an animal required to be registered under these regulations shall submit an application for a certificate of registration to Animal Control on the form provided by Animal Control or a legible copy of that form. An animal may not be transported to a location where it is required to be registered until the registration number and tag have been issued by Animal Control.
ANIMALS THAT ARE DANGEROUS AND IN NEED OF CONTROL IN HARRIS COUNTY, TEXAS
The following animals are found by the Commissioners Court of Harris County to be dangerous and in need of control in the unincorporated area of Harris County, Texas due to the animal’s size or nature, concerns of zoonotic disease transmission or other characteristics. This determination is made pursuant to the authority granted in Chapter 240 of the Texas Local Government Code, specifically including the authority granted pursuant to Section 240.002. The following “wild” animals are hereby determined to be dangerous and in need of control in Harris County, Texas pursuant to §240.001:
Non-human primates 
Houston only has a ban on monkeys over 20 pounds, so it is legal for you to keep a capuchin in Houston.
ARTICLE III. KEEPING OF WILD ANIMALS*
Sec. 6-51. Wild animal defined.
As used in this article, the term wild animal shall mean any mammal, amphibian, reptile or fowl of a species that is wild by nature and that, because of its size, vicious nature or other characteristics, is dangerous to human beings. Such animals shall include, but not be limited to, lions, tigers, leopards, panthers, bears, wolves, wolf-dog hybrids, cougars, coyotes, coyote-dog hybrids, raccoons, skunks (whether deodorized or not), apes, gorillas, monkeys of a species whose average adult weight is 20 pounds or more, foxes, elephants, rhinoceroses, alligators, crocodiles, caymans, fowl larger than a macaw, all forms of venomous reptiles and any snake that will grow to a length greater than eight feet. The term shall also include any animal listed as an "endangered species" under the federal Endangered Species Act of 1973, as amended, or any fowl protected by the federal Migratory Bird Treaty Act. The term wild animal shall not include gerbils, hamsters, guinea pigs, mice and domesticated rabbits.
(Ord. No. 99-404, § 1, 4-28-99)
Sec. 6-52. Possession prohibited.
(a) It shall be unlawful for any person to be in possession of a wild animal within the city.
(b) It shall be unlawful for the owner or other person in possession or control of any lot, tract or parcel of land within the city or any residence or business premises situated thereon to knowingly suffer or permit any other person to be in possession of a wild animal upon the property, residence or premises.
(c) As used in this section, the term to be in possession includes any harboring or keeping of a wild animal, whether on a temporary or permanent basis and includes, without limitation, holding or keeping the wild animal for the temporary and/or limited purpose of sale or transfer or offering of sale or transfer to another person.