Transcribing Calvin's Job sermons, encountering only "specially" and no "especially".
The difference, one is generally followed by a verb (past participle) and has no comparison, while the other is not the case, respectively.
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Which one is followed by a verb? "Specially" ??
That's a quick response. If you see the link I've proved, you will notice the use of specially is followed past participle verb: ie. specially prepared, specially minted, etc.
This is just my brief observation. I have not done any academic research on it.
On the other hand, they are also known to use interchangeable (especially & specially). So although their difference exists, it is often times ignored.