Antonalism is a term that relates specifically to music that has no key or "tonic."
Funk and Wagnall's Encyclopedia tells us that atonality is 'the absence of the system of harmonic relationships called tonality.'
eNotes.com provides a definition:
Atonality in its broadest sense describes music that lacks a tonal center, or key.
More simply put, atonality is the absence of tonality. Basically, music is generally written in a key, and whatever key is selected for a piece of music, within that key is a note referred to as the "tonic."
All tonal music has a key, such as "C major" or "F minor", etc. All the notes in a key are related to a central note, called the tonic.
The tonic is the center of the piece, and it is out of that tonic that the music grows, continually leading back to that tonic, and resolving itself at the end of the music—once again to that "tonic" or central note—providing closure. When one hears a song that does not resolve itself to the tonic, the music seems to hang unfinished in the air. This can leave the listener unsatisfied: it seems almost unnatural.
If the tonic is the center of a piece of music, "atonal" music does not have a tonic or a center to which it continually returns.
The music is not unfinished, but is structured in a different way. Tonal music concentrates on the central pitch or the tonic. Atonal music does not present a pitch that is more important than the others, leading the other notes. In atonal music, all the notes are equally important—none takes precedence over another. It may be difficult to listen to since our ears are trained to listen, unconsciously, for the tonic. It is simply a different manner of musical composition, implemented by composers such as Debussy and Ravel.