There are some very different opinions here, but also copy variations. Here is my view and I have both lenses and have shot with four different copies of the 85 f1.8.
The 85mm f1.8 is stunning value. If you have a decent copy and it is micro-adjusted fine, it is very sharp indeed from wide open. Very. It makes a superb landscape lens a few stops down also. Its only weakness is CA, which is pretty bad. Character? Very pleasant, but nothing to write home about. I can only describe my personal copy as a sensational general performer for the money. However.....
The 85mm f1.2 L II is a completely different beast. It is slower to focus, but I have always found it a total non issue for portraits as the lens is only having to keep up with very small movements of the subject. Portraits aint sports, unless you are dealing with animals or kids

Regarding image quality, mine is a very sharp lens indeed right from f1.2, where a face will fall within the frame. I could not care less about the edges and corners at f1.2 as this is only ever used for people. There is a gentleness to the rendering at f1.2, but this is most likely spherical aberration. The lens is still resolving plenty of detail. By f1.6 the contrast has picked up noticeably, vignetting reduced and it is a more technically 'clean'. But.... the look this lens produces is IMHO sensational. Bokeh is buttery way past f2.8 and even at f2.8-4 this lens has a look that is just great for people at all apertures. There are so many technically perfect lenses around now and I own some of them, but none come close to the big fat 85 1.2 L II for 'human' rendering in my view. My 85mm f1.8 EF doesn't and neither does my 85mm Zeiss Batis. This last lens is technically better at f1.8, with more contrast, more saturation etc, but its people pictures feel much less human and less organic.
Its up to the individual to decide how much these subtleties matter, but for me, I cannot think of a lens I would rather use for taking musicians' portraits than the 85 1.2 L II....
The 85mm f1.2 L II is lens that does not exist in any other system and probably never will. Poeple buy stats. People buy MTFs. Building a lens that is anything less than as perfect as possible is likely commercial suicide now. Look at the new Nikon 105 f1.4. Amazing performer, but hasn't got anywhere near the character of the Canon 85 1.2. I think its the best in production portrait lens in the world, but then again I have not used the Sony GM 85 1.4.