Tonality is fundamentally an aesthetic, mental procedure that has levels of skill.
Atonality, doesn't really exist as a "property" of music. You'll often find that composers who write music normally given this description, not being fond of the term.
I think we probably have different definitions or tonality/atonality. For me, tonal music is music that follows the conventions of the 'common practice period' - late baroque to romantic periods - where the music revolves around a 'tonic' note which is defined (more or less) by an authentic cadence. Atonal music, in that case, is simply music without a tonal center or tonic.
Tonality/atonality isn't a dichotomy either because you can have all sorts of music; tonal, atonal, polytonal, modal, polymodal, chromatic, polychromatic, symmetric, mircotonal, serial music, etc…
I'm curious to know in what sense you would say that twelve tone serial music isn't atonal when it was specifically designed to abandon the tonal center, making each of the twelve notes of the chromatic scale equally important.