Chastise means to punish or castigate. Chasten means to discipline or subdue. Chastisement is harsher, and chastening can be subtle and event gentle. Only chastisement would involve physical force or violence (though these are not essential to its meaning). Chastening usually comes in the form of verbal or social cues or formal rebuke.
Both verbs are rooted in the adjective chaste, and they were interchangeable until the early 19th century, when they began to differentiate. They have not fully differentiated, however, as they are still often used interchangeably, and mixing them up is not a serious error. If you can’t remember the difference between them or have trouble telling which one is most applicable, chastise is the safer choice, as it is the more common of the two and has a broader definition.
Chasten, the less common of the two, means to discipline or to subdue—for example:
With no kind of critical tradition to chasten him, his force is often misguided and his work shapeless. [The Danish History]
She seems more committed to small, common-sense government than any of them and could potentially rally the public to chasten a spendthrift Congress. [Toronto Sun]
Selling his city abroad was chastening, the mayor says: most Chinese had never heard of it. [The Economist]
And in reading these examples, note how those who are chastised are punished or harshly criticized rather than merely tamed or brought under control:
Many critiques of Badu chastise her for being nude around children. [Alternet]
He went on to chastise the use of RNC funds at a Los Angeles strip club and called it “the straw that broke the camel’s back.” [Seacoast Online]
The kneejerk response to this is to chastise the “Spanish system” as profoundly unequal and top-heavy. [Times Online]