Both forms of satire are subcategories of verbal irony, which focuses on the gap between what is said and what is meant. And both use this irony in comical ways with the intent to provoke laughter. By laughing at something incongruous, ridiculous, or foolish, the audience for satire performs a socially corrective function. The principle difference between Juvenalian and Horatian satire lies in the approach to corrections.
Juvenalian satire tends to be more caustic, biting, and even cruel. It is meant to purge society of elements that are dangerous to the collective well being, and sometimes this means banishing the inappropriate action, mocking him or her out of the community, or at least defeating this character or impulse. Often, anger and moral indignation lie behind the impulse to write Juvenalian satire. The Augustan era in England found several writers adept at this. Swift, often Pope, and Johnson used biting satire to comment on the folly or injustice of their day. In France, Voltaire's Candide has the same tone. Late night "news" comedy shows also use Juvenalian satire in a similar way. We might even think of Malvolio's treatment in Twelfth Night as Juvenalian in that Malvolio is mocked to the point of cruelty and eventually driven from Illyria's society, an apparently defeated man.
Horatian satire is more gentle. Here, the impulse is to show the audience their own folly and to purge it from the individual, rather than seeming to solidify around an external element. By seeing our own folly and laughing at it, we find in Horatian satire a correction to social ills as well, but the tone is corrective more than purgative. When Puck says "Lord, what fools these mortals be" (A Midsummer Night's Dream), we find an example of Horatian satire. Shakespeare typically will use irony to elicit laughter at human folly but with an impulse toward strengthening community through the realignment of individuals to social well being. This, again, is why Malvolio's treatment seems so shocking, as in Shakespearean and Horatian satire, we generally expect to find a place for everyone. Jane Austen also is generally more Horatian in her satire, suggesting characters are more misguided than actually dangerous.