It's strange how a single word can have so many different shades of meaning. Take the word "romantic." As one can easily see, it has as its root, Roman, meaning that having to do with ancient Rome...not modern Rome, that's Italian or Italianate. Thus, the fields of art, architecture, archaeology, literature, music, and history have first priority in the use of the term. Yet most people, in using the word romantic, think nothing about Rome, architecture, archaeology, or history. To most people it conjures up images of two people in love, dining out, walks along the beach, kissing, hugging and all that other mushy stuff. And if you should mention "feeling romantic," well, mushy can get downright...well, let's not go there.
In the fine arts, the word "romantic" has always been a nostalgic word, originally harkening back to Rome, but in its most recent incarnation, meaning melodramatically heroic, or sad, or lovesick--lots of incredibly handsome macho men rescuing fair damsels in distress from horrendously ugly ogres. In painting, it can be seen in the work of Benjamin West, Ingres, Goya, Géricault, Gros, and Delacroix. In sculpture, we find Jean Antoine Houdon and Antonio Canova; in architecture, Karl Langhan's Brandenburg Gate (Berlin), Piere Vignon's Church of the Madeleine (Paris), or Charles Garnier's Paris Opera. You'll have to ask someone else if you want to know about Romantic literature and music. Whatever the case, you'll find the major part of it having been created in the first half of the 1800s, as art and artists grew tired of the strict, iconoclastic rules of the Classical Revival and its even more restrained Greek Revival styles. Yet, Greek Revival is considered a Romantic style of architecture, along with Gothic Revival even though the two are in no way similar.
In this country, the Greek Revival period just happened to coincide with the building of most of the current state capitol buildings, so there's a wealth of free samples all over the country except perhaps in the far West. I won't go into a lot of details regarding the Greek Revival style largely because everything you see in Classical Revival is also present in the Greek Revival, only toned down somewhat. Roof slopes are less steep, porticoes less likely to be two-storied and if they are, often they are flat-roofed rather than gabled. Columns are more likely to be square than round and many times flattened against a wall. Cornices are usually more prominent than in the Classical Revival, and decorative touches, both inside and outside, fewer, more restrained, more rectangular, and more...well...Greek.
What makes this style hard to pinpoint is that there are so many manifestations of it...some might say bastardisations...as to make a proper Grecian's head spin. Also, unless it's labelled, and by someone who knows their labels, it can be devilishly hard to tell the difference between it and earlier Classical Revival structures. Moreover the transition from Roman-inspired Classical Revival to the Greek Revival was so gradual, and the cross-breeding of the two so common, one might be tempted to simply call it Classical-Greek Revival...except that the Greek Revival is considered "Romantic" and Classical Revival is not...God only knows why. Maybe the Greeks were more "romantic" than the Romans.