In the suspension bridge, a large cable is strung between two (or more) pillars, forming the primary load bearing structure. The cables experience tension from crossing the gap, the weight of the cable itself is the primary load. Smaller cables, or rods, are then suspended on the main cable, and used to support the load of the roadbed.
In the cable-stayed bridge, the pillars form the primary load bearing structure. Some use of a cantilever approach is often used for support of the roadbed near the pillars, but areas further from them are supported by cables running directly to the tops of the pillars. This had the disadvantage, compared to the suspension bridge, that the cables pull to the sides as opposed to directly up, requiring the roadbed to be stronger to resist these loads.
The cable-stay design has a particular sweet spot, it is used for spans shorter than what a suspension bridge would be used, and longer than cantilevers or other designs. This sweet spot exists because the suspension bridge requires considerably more cable overall, while a full cantilever would require considerably more material in the roadbed.
Examples of cable-stayed bridges