There are many differences and similarities between modern and ancient heroes. Modern heroes are different because of the way in which society has changed. Ancient and modern heroes are similar, because they are based on archetypes. These archetypes gives an author guidelines and specifications to use when writing, but most authors don't need to use them because it is natural to write a hero that way. It is natural to write heroes with these archetypes because we have grown up hearing and seeing the archetypes, without realizing.
The very large difference between ancient and modern heroes are the characteristics of the heroes themselves. While the old myths valued men who were strong, and came from a better upbringing than most. While modern heroes are mainly still men, there are examples of women being the hero, such as the Black Widow or Wonder Woman. Also, with the modern heroes, it does not matter who or where they are from. Anyone, with the right opportunities, can be a hero. An example is Captain America, he was from a lower class and was as skinny as a stick, but with the help of the super serum, a experimental serum created to turn men into super soldiers, became almost invincible, he survived 70 years of being frozen in ice.
Also the types of journeys that the hero goes on are different today. I Odysseus' story, he had with gods, where as in modern stories, heroes have to deal with aliens who think they are gods. For example, Stargate SG-1, where there is a race of little grey aliens who call themselves Asgardians, and like to meddle with peoples lives. For instance the main character gets kidnapped by the Asgardians and cloned and sent back as the clone.
Some similarities are because of the archetypes themselves. Such as the trickster or villain, or the Holy Grail. For example, in the Avengers, Loki is the trickster and the villain, and the Holy Grail is the Terasect. Another similarity is when a hero tries to save a certain area or people. In The Odyssey, Odysseus wanted to keep his family safe and return to them, while in Harry Potter, Potter just wanted to save the Wizarding Community.