Domestic vs. Wild
Domesticated goat breeds have been interacting with humans for thousands of years, and selective breeding and conditions of living alongside humans have altered them to distinct degrees from their wild cousins. For this reason, a domesticated goat that escapes can never truly be wild, but rather is "feral." Likewise, a wild goat kept in captivity is not domesticated, but simply in captivity.
Taxonomy and Nomenclature
The vast majority of goats are organized into the genus Capra, with only a very few exceptions --- one of which is the American mountain goat, which instead belongs to the genus Oreamnos. Scientific nomenclature categorizes the domestic goat --- Capra aegagrus hircus (sometimes shortened simply to Capra hircus) --- as a subspecies of the wild goat, also known as Capra aegagrus.
Behavior Differences
Wild goats are typically active in early morning and late evening, and females and young often live in small herds (typically fewer than 20 individuals) while males wander alone or in small groups and seek out and compete for females only during the mating season. Goats are typically skittish of human presence.
Domestic goats, which retain their herding instinct, are often kept in far larger herds, usually only comprising neutered males, females and their offspring, while intact males are kept separately. Activity is diurnal, with feeding and social activity primarily occurring over the day. Domestic goats can be taught to do a variety of tricks or practice other habitual behaviors and are often handled from an early age, making them less skittish around humans.
Health and Physiological Differences
Domestic and wild goats present a few physiological differences, one of the most significant of which is skeletal differences between males. In bucks, the bones of wild goats are often longer and thicker, so that wild males stand up to several inches taller and are also broader than their domesticated counterparts.
Domestic goats tend to outlive their wild cousins by several years, likely as a result of prevention of or treatment for exposure to various harsh conditions, parasites and illnesses. Domestication also typically ensures steady access to food and water, which wild goats are not always afforded. However, domesticated animals were interbred to create various breeds, and in some cases these breeding practices have resulted in the formation or facilitation of various genetic diseases (or tendencies toward genetic diseases) such as dysplasia.