When you destroy a Horcrux, you are not necessarily destroying the soul.
Destroying a Horcrux involves destroying the object beyond physical repair, at which point the soul inside "leaks out" and is destroyed.
A Horcrux is "the exact opposite of a human", as mentioned in the books1, and "depends upon its container to survive". This pretty much answers your question: Basilisk venom/Fiendfyre are tools to destroy the container, the fact that the soul gets destroyed is a secondary effect, if indeed the soul gets destroyed.
We've already answered the question, but we can dig deeper:
An interesting case to analyse would be when Harry is killed. The soul fragment in him isn't destroyed, rather it comes to the "afterlife" as well, though it is ill-equipped to survive there. Since Nagini's death counted as the "destruction" of a Horcrux in the typical sense, Harry's death does too. So, when you destroy a Horcrux, the soul fragment does pass on to the afterlife anyway. Just as a soul fragment.
1. Which I don't have with me right now, so if anyone can get the exact quote, it would be much appreciated :)
1. Exact quote:
"Because a Horcrux is the complete opposite of a human being.”
Seeing that Harry and Ron looked thoroughly confused, Hermione hurried on, “Look, if I picked up a sword right now, Ron, and ran you through with it, I wouldn’t damage your soul at all.”
“Which would be a real comfort to me, I’m sure,” said Ron. Harry laughed.
“It should be, actually! But my point is that whatever happens to your body, your soul will survive, untouched,” said Hermione. “But it’s the other way round with a Horcrux. The fragment of soul inside it depends on its container, its enchanted body, for survival. It can’t exist without it.”
-Harry Potter and the Deathly Hallows - the Ghoul in Pyjamas