"Choice" is such a nice word that everybody wants to have it on their side.
"Choice" is also a fuzzy word, which may be why Mitt Romney is willing to call himself a supporter of "school choice". In the strange language of education politics, "choice" sometimes means advocating the partial privatization of school systems through charter schools – which Romney supports. It can also indicate support for voucher programs, which is another thing altogether – and which Romney is said also to support.
Charter schools are constrained by the same laws and policies that, for example, prohibit public schools from endorsing religion. Vouchers, on the other hand, allow parents to use public money to pay for private, mostly religious schools that are largely unaccountable to the public. So, for example, a voucher school may use your taxpayer dollars to teach its students that the earth is 6,000 years old. And a number of such schools now do just that.
You don't have to be a constitutional scholar to get that using public money to fund religious schools violates the letter and spirit of the first amendment. Even the radical conservatives in today's Federalist Society would agree that the US constitution would not allow the government to cut a check to, say, the local mosque in exchange for supplying education to local schoolchildren. That is why they invented "vouchers": by pushing the "choice" to use government money to subsidize religion down to the parents, the government can fund religious schools while pretending that it is not.