This is, to a degree, a problem of terminology. The term "classical" refers to Graeco-Roman antiquity, a period of slightly over a thousand years, beginning in archaic Athens and ending with the fall of the Roman Empire. Of course, artists in antiquity didn't think of what they were doing as "classicism"; they were simply creating what was, for them, contemporary art.
The term "classicism" began to gain traction on the Renaissance, reflecting a return to the ideals of classical antiquity and rejection of medieval style and values. The term "Renaissance" itself is simply a French word meaning "rebirth"; what was being reborn was classical antiquity.
The movement generally termed "neoclassicism" developed in the eighteenth century. The prefix "neo-" means "new." In other words, this was a "new" form of classicism, differing from the earlier classicism of the Renaissance.
The main characteristic of Augustan classicism was the value placed on reason and restraint. It especially valued Horace and the poets of Augustan Rome. It believed in symmetry, harmony, and imitation of nature in art and saw this as compatible with imitation of the great masters of antiquity; Pope famously said:
But when t' examine ev'ry part he came,.
Nature and Homer were, he found, the same.