The Romantics view the world in a rather romantic kind of way. But, to give you a description that is actually useful: the Romantics viewed the world as a place full of beauty, wonder and mystery. It was a world where nature was to be highly valued, and human beings considered just a small part of something much bigger in the universe. Religion played a significant role in this worldview too.
Meanwhile, those in favour of the Enlightenment viewed the world in a different light. Everything happened for a reason, the world was governed by the laws of science. There was an answer behind everything, and everything could fit to the rules that humans created (for example, standardising the language, introducing the metric system etc). Religion didn't fit in so well.
Basically, those for the Enlightenment were for order and reason, and those for Romanticism were for emotion, disorder and mystery.
But I don't see why these things have to be mutually exclusive. Sure, science gives answers to things, but it also raises questions and creates more mystery. Language may be standardised, but amazing pieces of literature can still be produced. In my opinion, both of these views are extremes. The Romantics don't take things quite seriously enough and the Enlightenmentalists are far too stoic and unable to appreciate the beauty in the world. So I believe that there is a golden mean between the two, Romantenment as I call it for a little joke, and which I believe to be superior to the other worldviews.