“Neoclassical” era refers to a literary revisiting of Greek and Roman culture; the ideals of order, restraint, accuracy, decorum, reason, while Romanticism found inspiration in Nature, human emotions, and personal inspiration. A good representative of neo-classical literature is Alexander Pope (but there can be found example much earlier, even before the Restoration–for example, Ben Jonson). With Romanticism (ushered in by the Lyrical Ballads of Wordsworth and Coleridge, but Blake came earlier), order and reason gave way to intuition, a connection to Nature, and concentration on emotion: subjective, spontaneous, nonconformist utterances demonstrating an interest in the connections between Man and Nature.
The most telling differences are mental attitudes vs. subjective emotions; and finding ideals in the past vs. finding perfection in Nature. Look at Alexander Pope–rhymed couplets–irony, political satire, aphorisms, intellectual wit, etc.
A little knowledge is a dangerous thing;
Drink deep, or taste not of the Pierian spring”
vs. William Wordsworth’s
My heart leaps up when I behold/A rainbow in the sky
So was it when I first began; so is it now I am a man”
The essential difference is here.