Neo Classicism:

From about 1920 to 1950, many composers including Igor Stravinsky and Paul Hindemith reflected a movement known as the neoclassicism. This period is marked by emotional restraint, balance and clarity.
Composers during this era turned away from gigantic program music and orchestral works. they preferred absolute music for chamber groups. Most of them modelled their pieces over Bach's music using clear polyphony. Theywrote fugues, dance suites and concerti grossi.
Most music are tonal and they use major and minor scales. Neoclassicism is more of an attitude rather than a style.
Expressionism:

Much of the music in the 20th century reflects an artistic movement called expressionism, which stressed intense, subjective emotions. It was largely centered in Germany and Austria in1905-1925 and gew out of emtional turbulence in the works of Wagner and late romantics like Strauss and Mahler. Expressioist painters, writers and composers explored inner feelings rather than depicting outward appearance. They used deliberate distortions to assault and shock their audience to communicate tension and anguish. In a reaction against impressionism, they rejected conventional prettiness.
Expressionist painters such as Kirchner, Nolde, Munch and Kokoschka often used jarring colours and grotesque shapes. Art also tends to be fragmentary.
Artists during that time were creative in many areas. Wassily Kadinsky was a painter, a poet, and a playwright while Arnold Shoenberg was both a composer and a painter.