This is an oversimplification, but the main difference between Neoclassicism and Romanticism is that the former emphasized structure, objectivity, and restraint, while the latter emphasized imagination, subjectivity, and emotion.
Neoclassical poets such as Alexander Pope admired the Classical Age and therefore tried to emulate the features of that age. In Pope's An Essay on Criticism, he writes about the benefits of ordering and restraining creativity. When the inspiration hits, Pope argues the proper poet (or critic) will utilize control rather than let his or her imagination run wild:
Tis more to guide than spur the Muse's Steed;
Restrain his Fury, than provoke his Speed;
The winged Courser, like a gen'rous Horse,
Shows most true Mettle when you check his Course.
Romantic poets such as William Wordsworth emphasized the individual, subjective experience. Wordsworth was more concerned with imagination and emotion. In the preface to Lyrical Ballads, he famously wrote:
For all good poetry is the spontaneous overflow of powerful feelings: and though this be true, Poems to which any value can be attached were never produced on any variety of subjects but by a man who, being possessed of more than usual organic sensibility, has also thought long and deeply.
Although it is not always the case, Neoclassical poetry tends to be rooted in Classical forms and themes of structure and restraint. Romantic poetry tends to focus on the emotional and imaginative experience of the individual. Another way to say this is that the Neoclassical poet might say "this is how poetry should be" while the Romantic poet might say "this is what poetry means to me."