What advantages does markdown have over asciidoc?

Markdown is a TREND. There is no reason to switch to markdown if you're satisfied and confortable with asciidoc. Markdown is fragmented and quite limited in comparison to other markup languages such as asciidoc, txt2tags or reStructuredText.

Markdown has some critical mass, and the grammar is smaller than that of Asciidoc.

Markdown does have the drawback that if you want something more complicated you're out of luck, or into implementation-specific syntax that may not fit well whereas Asciidoc can scale up.

There is a g+ post by Dan Allen comparing them and prefering Asciidoc.