Let's start with drama. Drama is different from the other two forms in that a story is told through dialogue. There can be stage directions and information about how certain pieces of dialogue should be delivered; however, readers and audiences are not privy to the thoughts of characters unless those thoughts are spoken.
Prose (short stories, novels, etc.) can explore the internal thoughts and emotions of characters. Prose can also provide huge amounts of setting details to readers that dramas tend to ignore or minimize.
Poetry can be difficult to define. It can have rhythm and meter and rhyme, but it doesn't have to be in that format either. It can be a narrative about an event, and it can include dialogue as well. It tends to be a bit shorter of a literary form, but poems like "Rime of the Ancient Mariner" and "The Lady of Shalott" are quite long. Anecdotally, I had a professor one year that said "poetry is creating the most imagery or evoking the most emotion with the fewest amount of words." That can not be applied to every poem, but it does provide some decent direction about how poetry tends to focus on emotions and the senses of readers.
Regarding the similarities, each form delivers information to readers. Additionally, each form is going to incorporate themes. While it isn't guaranteed, all three forms are likely to use figurative language throughout as well.