The aversion to the word "voucher" by Democrats has always been interesting to me. If you call it a "stamp" or "coupon" and apply it to food or medical care, they don't consider the idea to be of Satanic origins.- Comment at Cafe Hayek
It's not really about vouchers per se.
It's true that when people lack food or shelter, liberals have no problem with issuing government vouchers that pay private businesses to supply them. But when individuals lack educational opportunity, the issuance of a voucher, all of a sudden, becomes a potential gash in the national fabric.
The difference is that the food stamp and Section 8 vouchers increase dependency on the government. In the case of schooling, the dependency has already been established.