A paradox is when an author uses apparently contradictory statements to get at some underlying truth. A very famous paradox from a popular work of literature, Animal Farm, might be the pigs' revision of the Seventh Commandment, "All animals are equal," to read "All animals are equal, but some animals are more equal than others." On the face of it, this is obviously an absurd proposition. Nothing can be more equal than something else. Thus it is a paradox. But the author uses this paradoxical statement to show how one group of animals, the pigs, have subverted the good intentions of a revolution intended to bring about equality to their own ends.
Juxtaposition, on the other hand, simply refers to placing two contrasting elements in proximity to each other to achieve some literary effect. Shakespeare, for example, would often include a comic figure in the midst of the darkest, most dramatic sections of his plays. The porter in Macbeth's castle is a good example of this, making ribald jokes as the audience is still reeling from Duncan's murder. In other works, we see juxtaposition of imagery, such as the light of civilization and the darkness of savagery in Heart of Darkness.
Juxtaposition is, according to the dictionary, the fact of two things being seen or placed close together for the purpose of contrast. A paradox is a statement that despite reasoning leads to a conclusion that seems logically unacceptable or self-contradictory.