In my opinion most of the time there's little or no difference between these two words.
I interpret the words in the following way:
special: it means particular,distinguished in a distinct way, or even designed for a particular purpose
especial: In my eyes very uncommon, the word especially is often heard.It means "noteworthy or particular"
(2) Maybe the best way to get some idea about the difference is to study their adverbial forms:
(a) I went to Japan specially to learn the language. = I think that it means that my specific goal for going was to learn Japanese.
(b) I went to Japan especially to learn the language. = I think that means that, yes, my main reason was to study Japanese, but other reasons included the chance to see Mt. Fujiyama, ride the bullet train, etc.
(3) At this moment, I am looking at The King's English written in 1906 by the Fowler brothers. They wished to thank someone for his help. They write: The especial thanks of the compilers (the writers) are offered to Dr. ...
(4) Mr. Bryan Garner, an American expert, says "especial" is the opposite of "ordinary."
(a) He gives this example (I have reworded it): The media are entitled to especial rights and privileges.
(b) According to Mr. Garner, there's a difference between (my examples) "a movie of especial power" and "movie of special power."
The first = a movie that has something very distinctive or significant about it; the second = a movie that has superhuman or even power from another planet. I don't think we would want to express the meaning of the second example.
Let's see what other Forum contributors can teach us.
People are waiting to help.
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