Both epic heroes and Romantic heroes are usually the main protagonists in literary works. An epic hero is from history, a myth, or a legend. He is often somehow connected to a god (or is often even the son of a god, such as Zeus), although he lives his life among mortals. Note that epic heroes do not have godlike power, but they are smarter, braver, and stronger than the average person. The epic hero is usually on a quest or voyage, where he faces trials and adversaries that change him in ways that reflect the morals and values of the society from which he came. Some examples of epic heroes are Achilles, Perseus, Hercules, Odysseus, and Beowulf.
The Romantic hero emerged in the literature of the Romance period of the 18th century. The Romantic hero is less conventional than the epic hero. Many Romantic heroes have been somehow rejected by society or are otherwise non-conventional in their ideas and ways of life - some might be recluses, some might be obsessed with a lost love. Like epic heroes, Romantic heroes are also often on some kind of quest, either a physical quest or an emotional/spiritual one — but many times the Romantic hero's quest begins from a desire to fulfill something for himself and ends up serving a greater cause. They are often innocent but intuitive, and more alienated or disillusioned than the epic hero.
Some examples of Romantic heroes would be Hawkeye from Last of the Mohicans and Victor Frankenstein of Frankenstein. Some modern examples of a modern Romantic hero are Ponyboy from The Outsiders and Harry Potter.