1. All twelve chromatic tones must be used equally
2. Melodies are built based on tone rows involving all twelve tones
3. The matrix may be used to find inversions and other versions of the original tone row, to provide variety.
Atonal music, by contrast, does not have to use all twelve chromatic notes or follow such a mathematical structure. The compositional process would then be much more intuitive.
If you haven't figured out yet, I'm fond of definitions, so here's one I found for Serial Music/ The Twelve Tone System:
Twelve Note Method / Dodecaphony :
"Serial music is constructed according to the principle, described independently by Hauer and Schönberg in the early 1920s, of 12-note composition. According to the Schönbergian principle, the 12 notes of the equal-tempered scale are arranged in a particular order, a series or row, that serves as the basis of the composition. Serial technique requires that the succession of notes be ordered as they are in the row, but simultaneities--chords--have no succession within them, so the principle of order relations does not apply to them. In Schönberg's Method of Composing with Twelve Notes Which are Related Only to One Another, the note-row may be used in its original form, or inverted, or retrograde, or retrograde inverted; in each of these forms it may be transposed to any pitch (each note-row may thus have forty-eight possible forms). All the music of the composition is constructed from this basic material; any particular note may be repeated, but the order must be maintained. Octave transpositions are permitted. Notes may occur in any voice, may be used either melodically (horizontally) or harmonically (vertically) but the entire sequence must be employed before the row may be repeated. The row is normally designed to avoid outlining the triads or patterns associated with tonality. Berg's Lulu (1937) and Schönberg's Moses und Aaron (1957), both written entirely in serial technique, are considered the two masterpieces of the serial repertoire. Both employ the same arduous vocal style as Wozzeck and are significant undertakings for performers and audience alike. Later developments of 12-note theory introduced the idea of using six-, four- or three-note segments of a row as compositional elements. As originally designed by Schönberg, the method was intended to preclude tonality, though later composers, notably Berg, found ways of using the technique in a tonal context - as indeed did Schönberg himself."(taken from here )
And here's a definition I found for Atonality:
"(English, German, Spanish) music that avoids a key centre but is not constructed on serial principles, such works being written generally without a key signature."
(taken from here )
You can also visit this forum and the Chicago Symphony Orchestra for more information on atonality.
I hope this properly answered your question. I know it more than answered mine!