Hope is something you wish for, something you're optimistic about and you expect or want to happen. When a person does something they hope for success. When something (you're optimistic about) is happening, you hope for a good outcome.
A dream is something you desire, something wonderful, a fantasy. It's not based on reason. For example, you dream about all the things you want to do. Or you dream about all the places you could go. If want your dreams to come true, you need to get up and make them happen.