Why (so) sure? vs How (so) sure?

None of them are grammatical full sentences, if we don't consider "why so" as having a meaning of its own. And we'll do that in a minute, but let's first pretend that it doesn't have one.

Possible full sentences would be:

Why are you so sure?

How are you so sure?

Why is it that you are so sure?

How is it that you are so sure?

The forms you all have are eliding part of these sentences.

Now, those that do not contain so could be construed as elisions from different questions; asking why someone is sure of something is different to to asking why they are sure to the extent that they are. Of course, we could be asking that, but that's not the form that has become idiomatic, as we'll come to.

The way elision works with contexts is complicated, in that we can understand e.g. some headlines fine that we'd trip up on in the middle of a passage of prose.

And similarly, while "how so sure?" doesn't scan right, we still understand it, and it would unusual, but not utterly bizarre, for someone to use it in informal speech.

Likewise with "why so sure?" except that "why so" has been used so much as to become an idiom of its own, indeed one that is defined in its own right in at least one dictionary.

And so, after that little journey through elision, we can see that because "why so" has been so often used as to become an idiom, we can actually understand "why so sure" without reference to elision at all, but in terms of that idiomatic meaning.

("Why so" was once also used as an expression of relief or acquiescence, as it is used by Shakespeare and continued to be used into the 19th Century, this other meaning might or might not have helped "why so" in the sense here come into being).

"How so" has had a different journey toward becoming an idiom, in that it is often found as the entirety of a question ("How so?"). As such, just as we can understand "why so X?" as an idiomatic form with a defined meaning, so we can "how so?", but not "why so?" or "how so X?" because that is not what those idioms mean.

Why so sure works because the are you is understood to be elided from the familiar expressions: Why so blue? Why so serious? Why so sad/down/etc.

How so sure has no such similar counterparts, so sounds strange.

How sure, again, is short for how sure are you? The same structure can't be made with why sure. If you google "how sure", you will get many hits for "how sure are you(...), but not so for "why sure".

Broadly, there is no difference. All those options have the same meaning and here, "so" matters only for emphasis.

"Why so sure" works because, as in the similar expressions "Why so blue/down/happy/sad/etc" the "are you" is understood to be a contraction. That's not a lazy, modern thing. I'm sure it's seen in Shakespeare (1564-1616) and almost certain it's heard in Chaucer (1343-1400)

"How so sure…" has at least the same counterparts, though it's less likely.

"How" and "why" do keep different meanings but those differences truly do not matter here except for one example: "How sure…" is just as likely to stand for "How are you sure?" as for the wholly different "How sure are you?", which will never work with "Why…"

Otherwise too few speakers, including natives, understand or even recognise a difference.