Interestingly, both camelids and equines are native to North America - after having colonised Eurasia, both vanished from North America, most probably having been hunted to extinction after humans colonised the continent, roughly 8-10 000 years ago.

HORSES:

Horses returned in 1494, with the second expedition of Columbus (to Hispaniola at that time). Later explorers and colonists also took horses. In a very literal sense, they returned home, and lost or escaped animals quickly established a large population of feral horses. This time though they were appreciated by the natives as more than meat, and they quickly became an important part of indigenous culture.

The homecoming of the horses was so successful, in fact, that the US government started exterminating them in an organized manner, since with the decline of large predators, they quickly became pests. It's still a recurring problem actually.

CAMELS:

As for camels, they were also brought to North America, though much later, only when arid lands in the west became a major logistics issue. There have been attempts by the US army to use camels in pack trains, and about 75 animals were imported. The experiment was ultimately a failure for a variety of reasons: The Civil War started and the "experimental camels" were seized by confederate forces, and also the camels were reportedly treated very poorly by drivers who were opposed to using them. After the Civil War, the expanding railroad network delivered the last blow to the camel project. The animals have been sold to various places, from slaughterhouses to zoos. Not all the animals have been accounted for though, and some have clearly wandered off, but probably far too few to start a population. There are rumours of sightings, though :) .

EDIT: To give you a short answer, I think above information indicates that horses and mules dominated the field mostly because they were there first, had a huge population advantage, they were accepted by people of predominantly european heritage, and because the people who built business on them were wary of any kind of competition and replacement.

Sources:

Stephen Budiansky - The Nature of Horses

Smithsonian.com