Meine Arbeit ist getan.

My work is done.Would it make a difference if instead of “getan” it was “fertig” in the meaning of this sentence?

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Apart from it being pretty much a set phrase, I don’t think “fertig” would work here and retain the same meaning.

When something is “fertig”, you have a concrete result that you can do something with. Like when you have written a paper or built something.
So because of this, if you do use “fertig” here, it would change the meaning so that “Arbeit” no longer means “job” in general, but for example a homework, a thesis, or an essay, i.e. some kind of paper that you turn in and get a grade for.

The same double-meaning applies to “Aufgabe” (= task).

The lines aren’t as clear-cut as I depicted here. Given the right context, you could use “fertig” in the same sense, but without context, you would probably invoke the wrong associations in the listener’s mind.
If I were to use “fertig” here, I would rephrase it to “Ich bin fertig mit meiner Arbeit.”. It is still borderline, but a little less so, because it puts the focus on the completion and less on what has been done.

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Pitti’s explanation is very good; it is indeed hard to pin down the difference.
I would like to add just one small thing:
“Getan” is passive of “tun”, so the focus is on that, what I have done.
“Fertig” means finsihed, so the focus in on the work.

Consider this example,
“Meine Arbeit ist getan. Nun müssen andere weitermachen.”
“My job is done. Now others have to carry on.”

In this case you never would say “fertig” . And I (not an English native speaker) would also not say “finished” here.

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