classicism is literary genre popular from the 16th to the 17th centuries and is heavily inspired by Roman and Greek culture. The genre is heavily riddled with an emphasis on society.
Neoclassicism is the literary movement which directly followed classicism beginning in the 18th century. It is popularly referred to as the "art of the ideal" and also expanded it's genre into art, architecture and music.
Neoclassical and classical writers saw themselves, as well as their readers and characters, above all as members of society.
The biggest difference between the two is the grammar and language used with each movement. Classic authors often riddled their works with figures of speech and highfalutin words that they often made up themselves. Neoclassic writers regularized grammar and vocabulary. The basic and less obscure became the trend.