I found this thread on wordreference.com in which it is claimed that "mimic" has frivolous connotations and "copy" has derogatory connotations. Specifically I am wondering how this distinction may relate to interpretations of Wittgenstein. If one can mimic another's gesture or expression, can one also copy it, and vice-versa?
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To me, copying involves a mechanistic approach, such as constructing the copied outcome from the source's details by following a systematic process. Mimicking, however, entails a convergent approach, where the mimicked outcome emerges from the usage of a more experimental or trial-and-error method.
This is compatible with others' claim that you cannot copy somebody else's voice, you can only mimic it; that is, you try to imitate and perfect the result it until you get the desired outcome. But you can copy somebody elses's notes; that is, you get access to the source and mechanically reproduce it piece by piece.
As an aside, mimicry is the technique used by animals (or plants) to "copy" each other's appearance or behaviour.
The OED has that mimic means to imitate or copy (a person, action, etc.) esp. for the purposes of ridicule or satire, or to entertain. Thus, I would say that mimic has a different connotation than copy, but has no formal difference in meaning when referring to persons, actions, etc.
Of course, as the OED points out, one wouldn't mimic a thing, one would copy a thing, i.e., the difference in meaning and usage rests in the thing being copied or mimicked. One could both copy and mimic a person's speech, but one could only copy a person's notes (as abhiii5459 correctly pointed out).
The difference lies primarily in their usage in real life.
You can mimic somebody's voice;you can't copy it.
You can copy somebody's notes.
A related term to consider in context would be 'imitate'.