In some cases, it would be more appropriate to say Neoclassical shifted to Romanticism, but there certainly were antithetical moves there.

Neoclassical poetry is characterized by objectivity, restraint, order, reason and at times, with the intent to reinvent past classical forms; not to copy, but to use their example to create new forms of poetry based on those concepts and even content. Language should be beautiful, restrained passion and it should reflect the ideal of society.

“Two principles in human nature reign;
Self-love, to urge; to reason, and to restrain; (A. Pope, Essay on Man: Epistle II).

Romantic poetry is characterized by subjectivity, emotion, melancholy and nature and as Wordsworth called it, "the spontaneous overflow of emotion reflected in tranquility." Language should be real, emotionally free and reflect the ordinary person and his/her engagement with nature.

“There was a time when meadow, grove and stream;
The Earth and every common sight,
To me did seem,
Appareled in celestial light.” (Wordsworth, Ode on Intimations of Immortality from Reflections from Early Childhood.  

This makes it sound like Neoclassical is all stuffy and Romanticism is a free spirit. To a large extent, this is true. But Romanticism was not just a reaction to the restraint of the individual in neoclassical literature. Romanticism was a reaction to the Historical moment of industrialization and the disappearance of the individual in capitalist society. Hence, the Romantics focus on nature. So, the antithesis is more between Romanticism and historical movement because there are examples of neoclassical poetry concerning nature and freedom and the focus on mankind (humanism) and the individual; neoclassical may treat these themes but it is just more characteristically about reason than emotion.