They are both satyrists of social issues which they wrote in their stories as mockery of the snobbery, hypocrisy and falsehood of the upper classes, especially those who are "in charge" of society.
Goldsmith has personified characters with characteristics that you would think are meant to evoke respect and love for the character only to find out that he was mocking the character and evoking laughter in the reader. Swift does the same in Oliver Twist, for instance, where he particularly pounds on the Laputians, who are the representation of all that is wrong in being human.
The contrast is mainly their focus. Swift is more of a humanist=-- he criticizes society from the point of view of how man has ruined himself, and society along with it. Goldsmith is more of a social observer, and writes and mocks about it as a whole.
While Oliver Goldsmith and Jonathan Swift were both satirists, they targeted different aspects of society in their works. Goldsmith, like Swift, mocks the society of his day, but his depiction of characters hinges on their representing types of people. They do not appear to the reader to be individuals in their own right - a method other writers such as Moliere uses. As such, much of Goldsmith's satirical focus is on society as a whole. In addition, unlike much of Swift's satirical work, one of Goldsmith's primary motivations was to make people laugh, often making his ridicule more subtle and easier for the reader to swallow. The gentle nature of Goldsmith's satire is puzzling, because it is often difficult to discern what is truly sentimental and what is truly satirical.
Jonathan Swift, on the other hand, does not run up against the same problem. In his works, his readers become very aware of Swift's satirical targets. In "A Modest Proposal"(1729) perhaps the most well-known of Swift's satires outside of Gulliver's Travels (1726), he targets not only the Irish who are lazy and apathetic toward their own lives but also the English who have put the Irish in such a situation. The very subject matter of the pamphlet alerts the reader to the nature of the work. Gulliver's Travels, like "A Modest Proposal," targets more specific aspects of society than much of Goldsmith's works. In his novel, Swift takes the scientific community to task, as well as humanity as a whole. As such, his satirical work tends to be much more pointed in drawing out its targets and more stinging in his indictment of them.