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The accepted answer says
The present-day difference doesn't seem to have any clear cause.
Which is the answer to the Why question of the OP: 'Why is "medicine" pronounced differently?'
Such is the situation often caused by asking why questions. All the other information in the accepted answer, as swell as it is, does not actually answer the question as to why.
Thus, the answer itself is in jeopardy of receiving a downvote because it doesn't answer the question why except to say we don't know why.
I have not voted on either the question or answer. It seems a good strategy if users try to reword their questions so they don't ask why. Perhaps even is there a reason for... is better than a flat out why question. At least in this case the answer there is no apparent reason is an actual answer to the question.
As Dan Bron comments,
As English is not designed, a lot of answers to “why” questions are “because that’s how the chips fell, that’s all; there is no rationale beyond ‘a series of accidents of history’.