Both of these famous authors worked principally in the 18th century, and both are famed for their satire and their incredibly comic writings. The difference between them, arguably, lies in two areas. Firstly, Goldsmith now is more famed for his dramas and poetry, with such plays as She Stoops to Conquer being synonymous with the author, whereas Swift is famous for his prose, and works such as Gulliver's Travels and "A Modest Proposal" are famously associated with his name.
In all of these works, satire is a key theme, and yet there does seem to be a difference in the kind of satire that is presented. Swift deliberately wrote satirical works with the chief intention of provoking change, as his excellent "A Modest Proposal" indicates. Swift wrote this treatise out of his outrage at the reluctance of the British government to do anything to alleviate the tremendous suffering of the Irish during a time of famine and want, and so proposed a scheme that indicated the monstrosity of the situation and exposed the uncaring, callous nature of the British:
I have been assured by a very knowing American of my acquaintance in London that a young healthy child well nursed is at a year old a most delicious, nourishing, and wholesome food.
His "modest proposal" to rear Irish babies as a food source to solve the famine is deliberately shocking and provoking. Swift makes himself monstrous in order to expose the monstrosity of others and in the hope of provoking social change. Goldsmith's works can be viewed more as being comedies of manners, with the situation in She Stoops to Conquer being very comic and hilarious, even though there is evidence to suggest that this work does also point towards deep social inequalities about the way servingwomen were treated by well-to-do males. Nevertheless, Swift's work brings with it a definite sense of provoking change in society, whereas Goldsmith's work merely comments on society.