In general, what makes modern history modern is the sorts of attitudes that people have held during the modern era. People of the modern era have generally put more and more stock in science and rationality and less and less on tradition and authority.
Up through the end of the Middle Ages, people generally did not hold modern attitudes. They believed things because that was what the Church said or that was what Aristotle said. There was not a general sense that things should be proven scientifically. Since the beginning of modern times, people have come to have a more modern attitude in which science and scientific proof are privileged over things like the authority of traditional sources.
Modern History is so called because it is relatable to our lives to a certain extent. The issues that mark modern history are still not dismissed as simply "past" but are still slightly prevailant and discussed in the modern world. For an example, take Aparthied- although that is modern history, it is an underlying issue still discussed even today.