They are not that different, are they? AsciiDoc looks just like an specific dialect of markdown.
Maybe a problem with AsciiDoc is that its name is scary. By the name "ascii" it seems to imply that there is a specific markup for typing non-ascii stuff, instead of simply typing the regular unicode characters (in whatever encoding).
Prevalence of Markdown is to a certain extent accidental. It is one of those many markup languages that are easy-enough to reimplement which helped with the spread and I think it was supported on Github early on.
Asciidoc is much more complex and so we only have 2 implementations around (a2x, asciidoctor).
I think it's an example of a phenomenon where there is a new-ish category of things, and that category is good, and there is one thing in that category that becomes well-known, so even though that thing is not the best in its category, it becomes wildly successful.
No, problem is the covered complexity. For Markdown I have never resort to any documentation - within its original scope. This is his big advantage and I appreciate this.
If you need more, you have almost certainly to look up things. This is true for AsciiDoc as well as for example ReST, the syntax of the former being a bit more appealing to me (just compare simple links for example) .
For this reason, I'm using a blogging system (Pelican) that supports all of them (including Pythons Markdown extensions). I'm completely free to apply the best suiting variant to the topics requirements.