"Politics are important."
Translation:Politik ist wichtig.
Garner’s Modern American Usage (3rd ed.) explains the difference this way: “Politics may be either singular or plural. Today it is more commonly singular than plural (politics is a dirty business), although formerly the opposite was true. As with similar –ics words denoting disciplines of academic and human endeavor, politics is treated as singular when it refers to the field itself (all politics is local) and as plural when it refers to a collective set of political stands (her politics were too mainstream for the party’s activists).”
Other sources, e.g., the American Heritage Dictionary (3rd, 4th, 5th ed.) agree. The usual tip-off that "politics" is going to be plural is a modifier such as his, her, and especially their or the. "The politics of the district are complicated."
"Die Politiken der Länder" is rare, but grammatically correct. You only would use it when talking about different political strategies.
fehredef, mizinamo et al., The OGD entry is: Politik die ~, ~en  politics sing., no art.; eine gemeinsame/neu ~: a common/new policy  (eine spezielle ~) policy; eine ~ der kleinen Schritte a gradualist policy  (Tatik) tactics pl.
The fact that there is a plural suggests that with definition 3 one could say "Ihre Politiken sind sehr unterschiedlich," meaning "Ihre Tatiken sind sehr unterschiedlich."
However, Duden online shows: "die Politik; Genitiv: der Politik, Plural: die Politiken […ˈliːtikn̩, auch: …ˈlɪt…] (Plural selten)" and provides no example of the use of the plural. All the examples are singular.
Q1. Can you or someone provide a correct use of "Politiken"?
Q2. Can someone offer a sentence in which "Politiken" is correct, and "Politik" would be wrong?
As already said (and Duden confirms this), the use of the plural is very rare. It is limited to talking about political strategies in comparison. So an example could be "Die Politiken dieser beiden Herrscherhäuser hätten unterschiedlicher nicht sein können". But even here one could have used the singular.
"But even here one could have used the singular."
"Die Politik dieser beiden Herrscherhäuser hätte unterschiedlicher nicht sein können." is just as good or better?
Wenn ja, werde ich immer den Singular verwenden.
Ach so! I could opt to keep things simple by always using the singular. Or I could indulge my native-English bias, and use the plural in this instance. It's ironic that the English translation uses a verb that is the same in the plural and singular: "The policies of these two ruling houses could not have been more different."