Per the Google Books ngram server, even in the British English corpus, quiet as a verb has long surpassed quieten in frequency. The preference for quiet as a verb is stronger in the American English corpus, but both forms are found in both corpora, and in both of them quiet as a verb is and has been the more frequent.
The OED uses quieten as a synonym in defining several senses of quiet as a verb—which suggests that there is no difference in meaning. It does not likewise use quiet as a verb in defining quieten (understandably using it rather as an adjective in that task), but according to its own entries quiet as a verb is longer established even in British usage than quieten.