Difference between "opacity" and "opaqueness"

The difference in use is that you would use opacity to refer to understanding, when something is difficult to understand or obscure.

opacity, n.

a. Obscurity of meaning; resistance to interpretation; impenetrability; an instance of this. rare before 20th cent.

or

opacity noun

the quality of lacking transparency or translucence: thinner paints need black added to increase opacity

the quality of being obscure in meaning: the difficulty and opacity in Barthes' texts

Opaqueness is the noun to the adjective opaque, and would be used to describe the extent to which something is opaque, or able to transmit light.

opaque

A. adj.

2.

a. Not transmitting light, not transparent or translucent; impenetrable to sight. Also fig.

or

Definition of opaque

not able to be seen through; not transparent: bottles filled with a pale opaque liquid

(especially of language) hard or impossible to understand: technical jargon that was opaque to her

Used figuratively, in an understanding = seeing the light sort of way, the two words can become very similar, as shown particularly by the Oxford Dictionaries entries.

Opacity also has several technical uses. As pointed out by several posters, it is used in programming. It is wrong to imagine that language exists only for programmers though - opacity also has a technical use in physics, and in linguistics, and in photography, and in the paint industry, and probably others that I am unaware of. For these localised uses it is probably better to ask in the appropriate forums.

"Opacity" is usually used in technical contexts, and invokes a nuance of it being a measurable quantity. "Opaqueness" is normally used when speaking colloquially, and can also be used metaphorically to mean "incomprehensibility resulting from obscurity of meaning" (Mirriam-Webster). To that end, it would sound strange to give an actual measurement of "opaqueness". For example:

This fluid has an ?opaqueness of 40 square meters per kg.

This sentence sounds strange.

This fluid has an opacity of 40 square meters per kg.

This is correct.

Not entirely a new answer; I'd like to modify the idea provided by @tchrist:

opaqueness — the state or fact of being opaque.
opacity — The degree of being opaque.

This material is not suitable due to its opaqueness.
This material is not suitable because of its high opacity.

I use opacity as a parameter between 0 and 100%, when describing the degree of transparency of something, most often web markup. For example, opacity = 0.2 would be very close to transparent, or "see through". A more typical value for my purposes is opacity = 0.8 which is only slightly transparent. Of course, a value of 1.0 or 100% means totally opaque, solid, not gauzy or translucent at all.

Opaqueness is more familiar to me as word used for prose communication, rather than webpage design. Although the word is legitimate, I don't find it easy to use in context. It seems more idiomatic to describe something as "more opaque" or "less opaque", although an example of usage of would be to say something had "an aspect of opaqueness to it".

Opacity is a measure of light blocking (non-penetration) by a material. It can go from 0% (transparent; no blockage) to 50% (translucent; partial blockage) to 100% (opaque - complete blockage).

By definition, opaque means light cannot penetrate (100% opacity). If light could even partially penetrate, it would be translucent, not opaque.

So opaqueness is the state or condition of being opaque (no light can pass through). Opacity is a measure of how much light cannot pass through.

I'd like to add this thought. While I heartily agree that opacity is highly preferable to opaqueness, I ended up on this page because I was concerned that the particular audience for which I'm writing at the moment might stumble over opacity or not catch the meaning at all, where the point of the sentence is to harshly criticize the addressee for deliberately making a process opaque for the purpose of concealing cheating. The concept has to leap off the page and smack the reader up side the head to achieve its greatest effect.

I've never thought about when "-icity" should be used and when "-ness" would be proper. Considering it now based on my own use and experience without any investigation, it seems to me that it may be as simple as whether 1) one of them grates on the ear, or b) adding "-icity" makes the noun too dissimilar from the adjective for the relation to be clear, or even to be recognizable at all. Consider opaque and unique. I may be mistaking familiarity for connection as to opaque and opacity, but unicity for unique definitely doesn't work.