The differences and similarities between the modern and ancient heroes are about the same. Modern heroes are different because of the way society has changed. Ancient and Modern heroes were similar because of the archetypes that are hard to change.
The differences between the two are many. A very large distance between ancient and modern heroes is the characteristics of the heroes themselves. The old myths used to liked Men that are strong and come from a nicer upbringing than most. While modern heroes are mostly still Men there are examples of Women being the Hero, such as The Black Widow or Wonder Woman. Also the modern heroes, it doesn't matter who or where their from, anyone, with the right chances, could be a hero. Look at Captain America, he was from a lower class and was as skinny as a stick, but with the help of the super serum became a super solider. Or maybe Bruce Banner/Hulk, he was a scientist and had an experiment go wrong.
Also, the types of journey that the heroes go on are different. Such as in Odysseus' stories he had to deal with Gods, in modern times we deal with Aliens who think they are Gods. An example would be From Stargate SG-1, there is a race of little grey aliens who call themselves Asgardians, and like to meddle.
Some similarities would be the archetypes themselves. Such as the trickster/villain, or the Holy Grail. An example would be in Avengers Loki is the trickster/villain, and the Holy Grail would be the Terasect. Another similarity would be trying to save a certain area or people. Such as in the Odyssey, Odysseus wanted to keep his family safe and return to them, while in Harry Potter it was just the Wizarding world.