Minor differences, weight, possibly stiffer sidewalls. You do need to be sure they are the same tyre, the name isn't always a reliable indicator, some Schwalbe models have different features and/or compounds between folding and rigid, the Mondial for example.
I have a few of wire bead tyres still in use, I'm in no hurry to change them, but when I am looking for tyres, the better ones do seem to be more likely to be folding.