I find it interesting that before the coining of the term atonality, the word tonal was not a word that was used to describe Western music from the Middle Ages until the first decades of the 20th century. The term "classical," when referring to music, is very broad. It's relative neo-form is just as broad, or even broader, considering the amount of water that has gone under the bridge since it was coined as a term.
Neoclassicism is a term that tends to refer, at least at this point and in a musical context, to works that take their inspiration from pieces in the "canon." Now, because of all of the previously-unknown composers who have come to light in this era of musical information sharing (particularly the on-line music libraries here, here, and here), we see that the old "canon" is only a drop in a huge bucket of the excellent "canon-quality" music that has wafted through rooms, concert halls, and houses of worship all over the world for the past 500 or so years.
Many of the 20th and 21st century words in our collective musical vocabulary are really just names for techniques. Minimalism is a technique that incorporates a lot of repetitive ostinato, and serial writing organizes its pitches in sets. 12-tone music uses rules to organize a series of 12 pitches in a way that avoids repetition (the opposite of minimalism, perhaps), but it is also possible to write music using rigid sets with fewer than 12 pitches. We also have many different ways of constructing and describing harmony: tertian harmony is based on stacks of thirds, and quartal harmony is based on stacks of fourths. Atonality is simply the absence of tonality (see above), and, though it is incorporated in a great deal of music, we have finally gotten to the point where its organized 12-tone form is no longer the "rule" when it comes to writing new music. It is very strange that the ear can tell the difference between polytonality and atonality, but I suppose our natural urge to organize music tonally helps us identify the necessary patterns, even if we do so only subconsciously.
Thinking in general about the musical seas that contemporary composers swim in, with species of every ethnic and popular variety pulling us this way and that, with electronic possibilities that boggle the mind (including keyboards that can divide the octave into as many equal or unequal parts as anyone could want), and with instrumental virtuosity like we have never seen before, we probably need a less relative term than "neo-classicist" to describe the current new-music practices.
When people ask me what kind of music I write, my usual response is that I write "enharmonic music." I say that it is "enharmonic" because I often need to figure out whether the "black key" pitches are sharps or flats, and it sometimes drives me batty. Because the music that I write tends to move rather chromatically from tonal center to tonal center, the handy-dandy key signature doesn't capture that many pitches for very long. The term "enharmonic" pushed its way to the front of my brain this morning (while I was listening to Grieg), and appeared in large letters as "n-harmonic music."
Yup. That's what I write: n-harmonic music. It might be what you write too. Perhaps it should be written as n-harmonic music. I don't pretend to understand the n-related math, but I imagine that there are many composers who do.