All of the words above are adverbs and can be translated as "again," and they are often interchangeable. However, they do have slightly different connotations, and this means that one adverb may be used in certain situations or expressions exclusively.

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The adverb wieder is used to indicate repetition or a return to a previous state, though it may be uncertain exactly when this will occur. The connotation is "back again" or "repeatedly." You would not say nochmal da, nor would you switch out wieder in the two other expressions below. 

Wir sind wieder da auf Ganovenjagd

We are back again, chasing hoodlums

Caption 24, Die Pfefferkörner: Abschied tut weh

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Das sehen wir in dieser Stadt immer wieder.

We see this again and again in this city.

Caption 13, Architektur: der Stadt Karlsruhe

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Wir sind jetzt leider schon wieder am Ende der Sendung,

Unfortunately, we are now once again at the end of the program,

Caption 20, Abenteuer und Sport: Fallschirmspringen

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A casual shortening of noch einmal (literally "still one time"), the adverb nochmal suggests an immediate or additional repetition. The connotation is "another time," "a second time," or "one more time." As you might guess, danke nochmal is a standard expression, and neither wieder nor erneut would be used here. 

Danke nochmal, dass ihr mir geholfen habt, aber ich muss los.

Thanks again for helping me, but I have to go.

Caption 32, Die Pfefferkörner: Alles auf Anfang

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Ich wünschte, wir könnten uns heute nochmal begegnen.

I wish we could meet each other again today.

Caption 3, LOTTE: Zu jung

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Then there is erneut, which indicates "once again" and can have the connotation of a fresh attempt or starting over. You can remember this simply by the fact that the word neu is contained in the word, or by the related verb erneuern ("to renew," "to refresh"). Erneut is also slightly more formal than the other two adverbs, and can imply that something is inevitable or expected to repeat. 

Erneut rüsten sich die Murmeltiere für den Winter.

Once again, the marmots prepare for winter.

Caption 9, Wilde Schweiz: Sommer

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Die Prinzessin bat ihren Vater erneut um Rat.

The Princess asked her father for advice again.

Caption 65, Märchen - Sagenhaft Der Froschkönig

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Aber die Königin starb bald darauf und der König heiratete erneut.

But the Queen died soon afterwards and the king married again.

Caption 11, Märchen - Sagenhaft Schneewittchen

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Further Learning
You'll find many more examples to compare on Yabla German.