
anders
differently
In a different, not identical way.
Ich stelle mir meine Zukunft ganz anders vor.
I imagine my future completely differently.
different / differently
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differently
In a different, not identical way.
Ich stelle mir meine Zukunft ganz anders vor.
I imagine my future completely differently.

different
Describes a difference in quality or character.
Seit dem Unfall ist er ein anderer Mensch.
Since the accident, he has been a different person.

unlike
A fixed construction that expresses a comparison or contrast; it means 'in contrast to'.
Anders als gestern scheint heute die Sonne.
Unlike yesterday, the sun is shining today.

among other things
A fixed expression indicating that the thing mentioned is only one example from a larger set.
In seiner Freizeit liest er unter anderem historische Romane.
In his free time, he reads historical novels, among other things.
'Anders' (different / differently) has two main uses. As an adverb, it describes HOW something is done and remains unchanged. As an adjective before a noun, it describes WHAT kind of thing it is and is declined.
Diesmal machen wir es anders.
This time we'll do it differently.
To express a difference or contrast, the fixed construction 'anders als' (unlike / different from) is used. It introduces the comparison.
Anders als erwartet, war die Prüfung ziemlich einfach.
Unlike expected, the exam was quite easy.
The expression 'unter anderem' (among other things; abbreviation: u. a.) is used to name one example from a larger set. It means 'as one item among several'.
Im Museum gibt es unter anderem Gemälde aus dem 19. Jahrhundert zu sehen.
In the museum, there are paintings from the 19th century to see, among other things.
'Anders' (different / differently) is often used in fixed combinations to express a change or alternative. Typical examples are: 'ganz anders' (completely different), 'völlig anders' (totally different), 'irgendwie anders' (somehow different), or 'nichts anderes' (nothing else).
Seit er zurück ist, wirkt er irgendwie anders.
Since he came back, he seems somehow different.
new
For a method that replaces an existing one, „neu“ emphasizes its introduction as a change. „Anders“ emphasizes that the method does not match the previous one. Depending on the statement, both words can describe the same method.

Ab morgen verwenden wir eine andere Methode als bisher.
Starting tomorrow, we’re using a different method than before.
Ab morgen führen wir eine neue Methode ein.
Starting tomorrow, we’re introducing a new method.