I don't know, but I suspect that prostitution is illegal in Las Vegas because it would hurt the profits of the other vice (gambling). Most people do not like whores trolling for business in their presence, and no good neighborhood where middle-class law-abiding people live will have open solicitation going on. Drugs, gambling, and prostitution are all dangerous to stable family life, which is the basis for all civilization, including free civilization.
Since 2003, Las Vegas mayor Oscar Goodman has repeatedly stated that he favors legalization of prostitution in the city, perhaps turning East Fremont Street into a little Amsterdam. Goodman said there are pragmatic reasons to back legalized prostitution. Those include the acknowledgement that illegal prostitution is occurring and that brothels could provide safer, regulated and revenue-generating sex, he said.