Sorry if I left people hanging with the trolleywire vs. catenary article. Essentially, as Arcady explained in the comments of the last post, catenary suspension uses two or more wires to suspend the contact wire between poles. The term catenary comes from the curve created by the sagging of a wire or chain between two points. In overhead contact systems, the catenary curve is seen in the support wire while the contact wire is connected to it by suspenders.
Some versions can be pretty ugly such as New Jersey Transit's catenary:
Photo Courtesy HeritageTrolley.org
Here's a crossover in Charlotte and a long straight stretch.
Unfortunately this is sometimes the ugly that people think about when they think about overhead wires. While it's music to my lungs, many other people don't feel the same way.
The catenary differs from a simple trolley wire suspension in that there are more wires as well as generally more visual obstruction. The catenary is used for higher speed lines because the tension created in the wire is great enough to keep the pantograph from bouncing up against the wire which can cause great damage. Ultimately trolleywire is perfectly good for streetcars in cities and I would generally believe that it's probably the first consideration of designers of modern streetcar systems.
Below is a Combino Supra under trolley wire.
Some fun images from 1924's Electric Railway Handbook