There is no difference between afterward and afterwards. Neither is more correct or incorrect than the other, and both appear throughout the English-speaking world. North American writers tend to favor afterward, while English speakers from outside the U.S. and Canada tend to favor afterwards. But this is not a rule, and exceptions are easily found.
Every word ending in the directional suffix -ward has a parallel -wards form,1 and afterward and afterwards are just one of many of these -ward/-wards pairs. In modern usage, the -ward forms are more common in North America, and the -wards forms are more common outside North America. By and large, the words in each of these pairs are interchangeable with each other. There are a few exceptions—for example, forward has definitions it does not share with forwards—but afterward and afterwards are so far undifferentiated, so you are safe using the form that sounds better to you.
North American publications favor afterward—for example:
Afterward, Smith’s eyes were glassy as he recounted his decision to play. [New York Times]
And afterward, well, that’s one more person genuinely sorry – and not experiencing mixed emotions – to see you go. [Globe and Mail]
When asked his reaction to the scene unfolding around him afterward, Jayson Werth said, “We got a long way to go.” [Washington Post]
And publications from outside North America favor afterwards—for example:
Armed police believed Mark Duggan was raising a gun he had been hiding under his jacket when they shot him in taxi, but the weapon could not be found by officers afterwards. [Telegraph]
Afterwards, media trends went from “Romney won” and “silent Lehrer”, to “Obama fights back” and “Romney confidence soars”. [Irish Times]
He said he had received several phone calls afterwards about problems caused by the outage. [Southland Times]