They are the same dynasty ruled by the same family. The reason why there is the Western Han and the Eastern Han is because in between the two, a man named Wang Mang usurped power from the Western Han emperor and established the Xin Dynasty. After he was defeated and overthrown, the Han dynasty was reestablished in Luoyang (in the east), rather than in the old capital Chang'an (in the west), hence the name "Eastern Han". In some texts you might come across Former Han and Latter Han, which is the same as Western Han and Eastern Han.