Fluff wise :
A half-dragon is the son of a dragon and... something. It looks draconic since its birth and has immediate access to his powers. He probably was an orphan or at least had one of his parents disappear.

A Dragon Disciple has draconic ancestors but that can be as far as 10 generations away. Hence the fact it takes 10 levels to become half-dragon, as the power of their blood slowly awaken as they use arcane magic. Except that, they are perfectly normal members of their species.