Despite their relative closeness in time, Swift and Goldsmith, in short, are each representative of a distinct era in English literature. The first half of the eighteenth century has often been termed the "Age of Pope," since Alexander Pope, Swift's close friend, is by far the most important poet of the period from 1700 to 1744 (Pope's death). It's also often termed the neoclassical period or the Augustan Age, since Pope and Swift explicitly took as their models, respectively, Horace and Juvenal, the chief Latin satirists of Emperor Augustus's reign, which began the Roman Empire after the fall of the Republic. The forty years or so after Pope's death are often termed the Age of Johnson, since Samuel Johnson was the dominant literary figure in London during that time (the mid 1740s to the mid 1780s). Swift thus was active during the Age of Pope, and Goldsmith during the Age of Johnson.
Unlike Swift, Goldsmith was not really a satirist. He's often characterized as being remembered for one play (She Stoops to Conquer), one poem ("The Deserted Village"), and one novel (The Vicar of Wakefield). Only the first of these, She Stoops to Conquer, has anything in it that can be termed satiric, and it's an extremely mild and good-natured type of mockery in contrast to the trenchant and deadly serious satirical, though also comical, works of both Swift and Pope. Goldsmith was a major figure in the transition English literature was undergoing in the middle to late eighteenth century, from what we consider a "classical" orientation to a style that was at least a precursor to the Romantic movement. In both "The Deserted Village" and The Vicar of Wakefield,the overriding mood is gentle, nostalgic, and sentimental. Goldsmith tugs at the reader's heartstrings, presenting situations that not only Swift, but other earlier writers, such as Henry Fielding, would have exploited for rollicking humor, or used as a means to ridicule people or manners.
It's instructive, in the case of "The Deserted Village," to look at the issue of form vs. content. Goldsmith, like Pope and most of the English poets from Dryden up through Johnson, writes in rhymed iambic pentameter couplets. In Dryden and Pope, these are called "heroic couplets," partly because of the mock-heroic style of their poetry. In their verse, each couplet is a kind of self-contained encapsulation of a single, pointed idea, and there is an edginess, a tension, within the couplet itself, as well as in the way ideas are expressed in their poems overall. In Goldsmith, beautiful as his verse is, this kind of sharpness is lacking. "The Deserted Village" is a wistful piece about a way of life that was gradually fading away in the English countryside.
It's difficult to make direct comparisons between specific works by Swift and Goldsmith because the two did not use the same literary formats. Swift did write poetry, but, in spite of some pieces that are fascinating and highly amusing, they are occasional pieces and jeux d'esprit. He wrote no plays, and his novel, Gulliver's Travels, is less a novel in the modern sense and more of an extended commentary on human foibles (a parable). His greatest works are prose satires. If we wish to limit our discussion to prose works, we can see that Goldsmith, in The Vicar of Wakefield, wrote fiction that became highly influential both in its time and later on, prefiguring the work of Jane Austen and the Victorian writers. Its emotionalism was attractive to European writers, and it is referenced by Goethe in his novel The Sorrows of Young Werther. So Goldsmith, unlike Swift, was in tune with later trends and the beginning of Romanticism in literature.