|Neoclassicism in a drawing by John Flaxman, 1795|
Modernism had great successes in the early 20th century. Like all fashionable movements, it fractured into different schools, each emphasizing how new and different they were. Composers who resisted modernism were herded into a camp of their own and labeled "neoclassical" by analogy with all the other little compositional cliques. But they were just composers. Virtually all composers from the beginning of time have written music that stood in some sort of relationship with music of the past. For most of them it was music of the recent past because that was all they knew. But since the growth of the study of music history in the 19th century and especially since the development of recording technology in the 20th century, composers have been able to study and listen to music from almost everywhere --and when!
Since there were no models from Classical antiquity to imitate, neoclassicism in music looked back to the late 18th century and the Classical style of Haydn and Mozart. One of the earliest examples of 20th century neoclassicism is the "Classical" Symphony by Prokofiev, an evocation of the crisp aesthetic of a Haydn symphony:
But then the term started being used more and more widely until the list of composers extended from Busoni to Hindemith, from Villa-Lobos to Manuel de Falla. When Debussy was influenced by the music of the Javanese gamelan, we could call it "neo-gamelan" style music, could we not? It makes just as much sense as calling Stravinsky's use of Baroque themes in Pulcinella neoclassical.
The point I am trying to make is that making reference to particular musical styles, especially those of the past, is an age-old aesthetic practice. But in the 20th century, the label "neoclassical" took on an ideological tinge. It has even been used to characterize some composers as fascistic! Because of their occasional use of traditional dance forms, even Schoenberg and Berg have been called "neoclassical". The use of the term has become so promiscuous that only the most avant-garde of composers, like Boulez and Cage, have not been called neoclassical.
I understand some of the reasons, but using a single term so loosely and with little specifics is of little value and usually misleading. In virtually every piece of music there are aspects that resonate with other musics, present and past, while there are other aspects that are specific to the work. When Bruckner uses the style of a chorale in one of his symphonies, why don't we call that "neoclassical"? If you want to talk about the use of older styles, then you need to talk about particular things and avoid ambiguous labels like "neoclassical".
Here is Manuel de Falla's Concerto for Harpsichord and Five Instruments, a piece that has often been called "neoclassical":
UPDATE: Just to add to the confustion, "neoclassical" is a term also used to describe some trends in mid-18th century opera seria.