Finally, a bit of a subjective comparison. Below are a few samples showing the quality and texture of the two lens’ bokeh effect (the way in which it throws the background out of focus).

(Inconsequential aside: Bokeh is from the Japanese boke-aji, ボケ味, or "blur quality". According to Wikipedia, “boke” is also used in Japan to describe lack of mental clarity, or senility.)

Here are some shots in a typical short-focus still life setting. 

Click on any of the image thumbnails below to view the full resolution file. 

Canon EF 85mm f/1.2L II USM @ f/1.2

Canon EF 85mm f/1.2L II USM @ f/1.8

Canon EF 85mm f/1.8 USM @ f/1.8

Where you can see the most difference, however slight, is in the patterns of the thin branches in the upper right areas of the background. The f/1.8 lens give a very subtle, yet visible, linear feel to the shapes, almost like the details are out of registration, as you’d see on a printing press. Let’s look at some different samples.

Canon EF 85mm f/1.2L II USM @ f/1.2

Canon EF 85mm f/1.2L II USM @ f/1.8

Canon EF 85mm f/1.8 USM @ f/1.8

These shots are considerably more obvious in how they handle the bokeh effect, the most notable indication is the green traffic light in the center of the street horizon. At f/1.2, the circle of the light is considerably bigger than when the same lens is shot at f/1.8. The f/1.8 lens is smaller still.

Once again, the branches tell the story as well. Take a look at these details.

Canon EF 85mm f/1.2L II USM @ f/1.2	Canon EF 85mm f/1.2L II USM @ f/1.8	Canon EF 85mm f/1.8 USM @ f/1.8

In the case of the f/1.2 prime's exposure, the “circles of confusion” of the out of focus branches in the skyline are smoothed to an even tonality. Shot at f/1.8, we’re starting to pick up some detail, but still not as much, and in not as disturbing a pattern as the f1.8 lens. Often the Bokeh Effect of Canon's L-Series 85mm f/1.2 is described as “silky”.