Wire beads (non folding) are a heavier bead that is (obviously) less flexible. For this reason, they are built into lower grade tires typically. The shape of these tires is typically much more rigid, less supple, and they tend to be cheaper tires. There are some "flatproof" tires with wire beads, but again they are not the best ride. I personally ride a non-folding ultra flatproof tire in the winter, because being stuck with a flat when it is 20 degrees is terrible (Continental SuperSport Plus), and ride a nicer folding bead tire when the weather is nicer and roads tend to be cleaner.