Is there one word to describe the difference between wide angle lenses that have a circular distortion and wide angle lenses that produce images where perspective lines remain straight?
The two types of lenses you refer to are:
Rectilinear lenses produce more 'natural' looking images but tend to stretch features towards the edges of the frame, so some subjects, e.g. faces, look odd. But they work well for interiors and landscapes. Fisheye lenses are generally regarded as special effects lenses but Matt's excellent answer gives some more practical uses.
The terms are fisheye (circular distortion) and rectilinear (straight edges).
Fisheye lenses are often unfairly branded as "special effect" lenses by some photographers, due to their near ubiquitous use in skateboarding magazines in the 90s, and the ease at which you can create unusual images when trained on nearly any subject.
However fisheyes have sever unique and useful properties for photographers and can produce very natural looking images when used appropriately. Firstly fisheyes can be made wider than rectilinear lenses. Much wider. Rectilinear lenses top out at about 90 degree vertical field of view, fisheyes have been made with about 220 degree vertical field of view. That's more than 180 degrees, i.e. you can see behind you!
This makes them well suited to scientific applications, but also times when you are extremely restricted for space as a photographer, or when you want to capture a really wide field of view but don't have time/space to plonk down a tripod and shoot a panorama.
Another useful property is that fisheye lenses preserve the proportions of objects (but not their precise shape) whereas rectilinear lenses stretch the proportions of objects that appear toward the edges of the frame. A very wide rectilinear lens can give an uncomfortably distorted image because of this. Combined with the fact that any horizontal or vertical line passing through the centre of the lens wont be bent at all by the fisheye this effect can be exploited for nature/landscape photography.
The horizon is often the only straight line in natural scenes, so getting the horizon dead centre in the image with a fisheye can often yield an image that looks like it was shot rectilinear, but without any of the rectilinear squashing in the corners. This works very well for beachscapes in particular.
Lack of straight lines in the subject + horizon in centre = very natural image (probably moreso than if shot with an ultrawide rectilinear lens):
There's one more thing to add. A lens that is Rectilinear usually has distortion of some kind but a lens that is free of distortion is called aspherical often abbreviated to ASPH.