This is a question that doesn't have an easy answer because first, though much of Romanticism was opposed to the ideals of the Enlightenment, it was also in some ways, paradoxically, an outgrowth of it; and second, both the Enlightenment and Romanticism can be seen as movements that were made up of internally opposing or contradictory elements.

Enlightenment is the term generally given to the dominant Zeitgeist of the period from the late 1600s to the end of the 1700s. It was a set of ideas and ideals that stressed rationalism, optimism, and a wish to transcend the limitations on man that had been imposed in the past by superstition and religion. Philosophers, scientists, and people in all of the arts including literature, music and painting were all a part of this wave of modern thinking. It would be simplistic to regard this as a monolithic movement, however. Voltaire, for instance, is always considered a key figure in the Enlightenment. But so was the earlier philosopher and scientist Leibniz, whose ideas were ridiculed by Voltaire in Candide. Leibniz believed this was the "best of all possible worlds," a principle restated in Alexander Pope's Essay on Man. Voltaire was against organized religion. But Samuel Johnson (who also ridiculed the "Leibnitian" thinking in Pope's Essay on Man—in spite of his general admiration of Pope), also a major Enlightenment figure, was a devout member of the Church of England. These are merely a sampling of the diversity of thinking during that period.

The same degree of diversity of belief (and of aesthetics as well), if not a greater a degree, existed in the Romantic movement that followed the Enlightenment. At the end of the eighteenth century, the emphasis on rationalism, or reason as the principal guide in life and in art, was replaced by a new emphasis on emotion, on the personal feelings of the individual, and on the darker, more irrational impulses in human nature. The Age of Reason (which Thomas Paine entitled his book of 1794-95, though it was later remarked by G. K. Chesterton that the Age of Reason had already ended by that time) focused on man as a being in control of his destiny who could create a positive future, since he had presumably liberated himself from the destructive beliefs of the past. In the Romantic era, many began to re-focus their attention on the remote past and on the mythic symbols that expressed the primitive, non-rational side of man. This is seen in works as diverse as Coleridge's Rime of the Ancient Mariner, Byron's Childe Harold's Pilgrimage, and Keats's "La belle Dame sans Merci." In the Romantic mindset, optimism and pessimism co-exist, but in my view, the latter is dominant. Often literature takes the form of an adult fairy tale, as in the stories of the German writers Ludwig Tieck and E. T. A. Hoffmann. These are tales in which, as in myth and legend, man is often punished for overstepping the bounds of what is permitted to him by a dark, mystic force such as fate, which governs the universe and impedes his will.

The irony is that, in my view, none of the developments in Romanticism would have been possible without the largely anti-religious climate first produced by the Enlightenment. The introspection and the focus on the dark inner soul of the individual that are the hallmarks of the Romantic artist were an outgrowth, paradoxically, of the belief that each person is a liberated being free from the constraints of religion—a freedom which Voltaire, Rousseau, and other Enlightenment figures had promoted. And as with the Enlightenment, the Romantic movement contained its internal contradictions. Some writers, such as Wordsworth, wrote poetry in a deliberately plainer style approximating conversational speech, which was, to Wordsworth, a direct expression of the human soul that earlier "poetic diction" had compromised. But others, such as Keats, often deliberately used archaisms to convey an olden-time feel, a re-creation of the atmosphere of centuries before which fascinated Keats and many others. In artistic and philosophical movements, there is seldom uniformity of thought or technique, and there are no simple answers to what distinguishes one historical trend in the arts from another.

Romanticism was, in essence, a movement that rebelled against and defined itself in opposition to the Enlightenment. For the artists and philosophers of the Enlightenment, the ideal life was one governed by reason. Artists and poets strove for ideals of harmony, symmetry, and order, valuing meticulous craftsmanship and the classical tradition. Among philosophers, truth was discovered by a combination of reason and empirical research. The ideals of the period included a faith in human reason to understand the universe and resolve the problems of the world, expressed in the couplet by Alexander Pope:

Nature and Nature's laws lay hid in night:

God said, "Let Newton be!" and all was light

The Romantic movement emphasized the individual self and sentiment as opposed to reason and was more pessimistic in attitude, viewing the intellectual and artist as solitary geniuses rather than integral parts of a social system. Rather than valuing symmetry and harmony, the Romantics valued individuality, surprise, intensity of emotion, and expressiveness. They looked back to medieval (or "Gothic") models as much as to the Augustan tradition of Rome.

The ideals of these two intellectual movements were very different from one another.

The Enlightenment thinkers believed very strongly in rationality and science.  They believed that the natural world and even human behavior could be explained scientifically.  They even felt that they could use the scientific method to improve human society.

By contrast, the Romantics rejected the whole idea of reason and science.  They felt that a scientific worldview was cold and sterile.  They felt that science and material progress would rob people of their humanity (this is, for example, one of the major themes of Frankenstein, by Mary Shelley).  In place of reason, the Romantics exalted feelings and emotions.  They felt that intuition and emotions were important sources of knowledge. 

Thus, the ideals of the Romantics and the thinkers of the Enlightenment were very much opposed to one another.