Green beans, string beans, wax beans, and snap beans are all, essentially, the same thing. But different. See the differences, such as they are, below.
Green, Wax, String, or Snap Beans
Green beans, wax beans, string beans, or snap beans are long and rounded. Most are green, but some are yellow (as pictured here) or even purple (see below). Heirloom varieties may still have a fibrous "string" running down their sides, but most varieties for sale today have had that inconvenience bred out of them. Steamed Green Beans are delicious with just a pat of butter and a sprinkle of salt. They are also delicious when turned into pickles.
Haricots verts, French green beans, filet beans
These delicate green beans are very thin. They are usually green, but yellow varieties are out there, too. Many people consider them the best of the green beans, and they are priced accordingly.
Purple String Beans
Purple string beans are simply purple version of classic green beans or wax beans. They loose their purple color when cooked, so consider them for raw recipes or lightly steam them and dip them into ice water to preserve as much of their color as possible.
Romano beans are flat and wide and flavorful. Smaller ones tend to be more tender. Large ones will have more developed bean seeds inside. They require a bit more cooking, but have more flavor. Try them as Braised Green Beans to bring out their nutty sweet essence.
Sometimes called yard-long beans, these beans are, in fact, a completely different family of plant from green beans. They are similar in flavor and look (except for their length) to green beans, however, and can be cooked in the same ways. Look for long beans between 12 and 18 inches long for the best flavor and tender texture.