however / again
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however
Expresses a contrast, a different perspective, or a new aspect; in return, on the other hand.
Die einen lobten den Film, die anderen wiederum fanden ihn langweilig.
Some praised the film, while others, however, found it boring.
again
Again, once more. This meaning is somewhat more formal or old-fashioned.
Der Antrag wurde abgelehnt, woraufhin der Anwalt ihn wiederum einreichte.
The application was rejected, whereupon the lawyer submitted it again.
As an adverb, 'wiederum' (however/again) can stand in different positions in the sentence. If it is in position 1, the verb follows. Often, however, it also stands directly after the subject it refers to.
Wiederum zeigte sich, dass die Planung fehlerhaft war.
Again, it became clear that the planning was flawed.
In the meaning 'andererseits' (on the other hand) or 'im Gegenzug' (in return), 'wiederum' (however) belongs more to written or formal language. It helps clearly separate arguments or perspectives from one another.
Das Hotel war teuer, bot aber wiederum einen exzellenten Service.
The hotel was expensive, but on the other hand it offered excellent service.
The meaning 'erneut' (again) or 'noch einmal' (once more) is rare for 'wiederum' (again) today and is considered old-fashioned or very formal. It is hardly used in everyday language anymore.
Nachdem der König seine Rede beendet hatte, sprach wiederum der Kanzler.
After the king had finished his speech, the chancellor spoke again.
again / back
Wieder and wiederum can both say that something happens again. Wieder is the usual word for this in everyday German today, while wiederum can sound more formal or old-fashioned in this meaning. Normally, use wieder; wiederum is suitable only when this particular style is chosen deliberately.
Nach der Reparatur ist die Maschine wiederum ausgefallen.
The machine broke down again after the repair.
Nach der Reparatur ist die Maschine wieder ausgefallen.
The machine broke down again after the repair.
