What is the difference between sapience and sophonce?

The words sentient and sapient have fairly clear meanings, but the neologism sophont appears to be a synonym of sapient. Yet in a number of fiction a kind of hierarchy is made between the three.

In a fair amount of science fiction, the word sentient is misused as a synonym of sapient. In fiction such as Star Wars, this is further mangled into a distinction between "non-sentient" or "sub-sentient," "semi-sentient" and "fully sentient."

Initially I thought the sentient, sapient and sophont distinction was just a semantically correct version of the Star Wars-style of terminology. Yet fiction that uses this "correct" terminology, such as Orion's Arm or The Eldraeverse, often has dramatically different definitions and ideas about cognition (such as sapience without sentience, which I have difficulty imagining).

Even sentience may even be quantified as a "sentience quotient," unrelated to sapience or sophone, on which both plants and animals are measured. I don't know where I heard the idea, but I was initially under the impressions that plants were not sentient at all (by lacking nervous systems) until the quotient measured their sentience as much lower than animals.

I have no idea which distinction between sapient and sophont is correct, if any.

There isn't any. "Sophont" is just another word used by some authors for the same thing. It appears to have originated with Poul Anderson's Trouble Twisters. See Where did the term "sophont" originate? for a discussion.