Opposite opinion vs Opposing opinion

Suppose I think that puppies are cute (which I do) and someone thinks otherwise (how dare they?), i. e. puppies are not cute. Is the latter the opposing opinion or the opposite opinion? If both adjectives fit, is there any difference between the two collocations?

For most ends and purposes, the two phrases are interchangeable. If there is a difference between them, it is that opposing opinions, because its form suggests an ongoing activity, is more likely to bring to mind the idea of the two opinions, or, more accurately, their proponents, being actively engaged in some form of a confrontation. Opposing opinions may thus be a particularly apt term to use for opinions that have been voiced, and defended against each other, in an actual debate (although there would still be nothing wrong with using opposite opinions in such a case). On the other hand, if the proponents of the opinions do not actively engage each other, and particularly if they do not even know about each other, then opposite opinions is a better term.

Here's one way of illustrating a significant difference between opposite and opposing when those words are applied to something like an "opinion" or "position". Note the relative difference in prevalence between the definite and indefinite article. First, here's opposing...

...and here's opposite...

Clearly there's a significant bias towards an opposing [view, whatever], whereas we normally use the definite article for the opposite. By implication, opposite is more specific (there's usually only one "opposite X", whereas there may be multiple "opposing X's"). To me, that means opposite is a stronger term.

Your scenario:

Person A: Puppies are cute. --> All puppies are cute.

Person B: Puppies are not cute. --> No puppy is cute.

Person B has offered an opinion that both opposes the opinion of Person A and is the logical opposite of the opinion of Person A.

A variation of your scenario:

Person A: Puppies are cute. --> All puppies are cute.

Person B: Not all puppies are cute. --> Only some puppies are cute.

In this case, Person B has offered an opinion that opposes the opinion of Person A but is not the logical opposite of the opinion of Person A.

Bottom line:

(1) There is only one opinion opposite to the opinion that “puppies are cute”, that being that “puppies are not cute.” Person B is offering the opposite opinion.

(2) There are at least two opinions that oppose the opinion that “puppies are cute,” those being that “puppies are not cute” and that “not all puppies are cute.”

Given this, you could characterize Person B’s opinion that “puppies are not cute” as either the opposite opinion to Person A’s or as an opposing opinion to Person A’s. You could use either opposing or opposite provided you use the appropriate article.

Addendum: In response to comment from @alekscooper:

Let Statement A1 be Person A's statement that puppies are cute. Let Statement B1 be Person B's statement that puppies are not cute. Let Statement B2 be Person B's statement that not all puppies are cute. Then we have the following:

Statements A1 and B1 are opposing opinions. Statement B1 is an opposing opinion to Statement A1.

Statements A1 and B1 are also opposite opinions. In addition to being an opposing opinion, Statement B1 is the opposite opinion to Statement A1.

Statements A1 and B2 are opposing opinions. Statement B2 is an opposing opinion to Statement A1.

Note: I answered this question because I am of the opinion that "all puppies are cute." I just got a Cavachon, which is a cross between a Cavalier King Charles Spaniel and a Bichon. Couldn't be cuter.