As the title says. Both tenses seem to make sense in this sentence.
If the rain had stopped sooner, I would have gone out for a walk.
To me, they look no different, and so do all the online examples. Can I distinguish them or are they essentially the same thing?
We use the term conditional to refer to forms such as I would go (which is also used to represent the future-in-the-past), and the term conditional perfect to refer to forms such as I would have gone (which is therefore the perfect form of the future-in-the-past).
Resources for English language learners use therm "third conditional" to describe this sort of sentence: the protasis (the if clause) contains a verb in the past perfect, while the apodosis (the main clause) contains a verb in the conditional perfect. For example, the British Council has this example:
If I’d known, I would have worn something nicer.
I think what you are trying to say (and you are correct about this) is that the conditional perfect has the same form regardless of whether it is seen as representing an unreal past (i.e. a hypothetical in which both the condition and the hypothetical result are now in the past) or a backshifted future perfect from "I will have worn".
In an if sentence it makes much more sense to see the conditional perfect as representing a hypothetical, but in some sentences it could represent a perfect future-in-the-past - for example, if today I said "by tomorrow I will have been here for ten days", then you could use reported speech to report my remarks (with backshifted tense) as: "He said that by tomorrow he would have been here for ten days."
Grammarians of modern English don't use the term "subjunctive" for this type of construction. The term "subjunctive" is now reserved for two forms (and even its applicability to these two forms is disputed):
Most would also agree that English has only two primary tenses (present and past). Some use the term "aspect" for the progressive and perfect forms; others use the term "secondary tense". Traditionally (and still in much modern pedagogy), forms such as "I will give" were known as "future tense", but today it is generally held that such compound forms should not be described as tenses.
