It is certainly the case that in usage these two adverbs are often confused and can sometimes be used with the same meaning.
specially - for a particular purpose
However, when specially is used to mean for a particular purpose, this form of the adverb is the norm:
- This shower gel is specially designed for people with sensitive skins.
- This computer programme is specially for children with learning difficulties.
- My father made this model aeroplane specially for me.
especially - particularly / above all
We tend to use especially for emphasis, meaning particularlyor above all:
- These butterflies are particularly noticeable in April and May, especially in these meadows.
- You'll enjoy playing tennis at our local club, especially on weekdays when it's not so busy.
Before adjectives, especially, meaning particularly, is more usual:
- The road between Cairo and Alexandria is especially dangerous at night.
- It is a bit nippy, but it's not especially cold for this time of year.
special - especial
Note that the adjective especial is rarely used nowadays. Its use is confined to particular contexts where it collocates with particular nouns, e.g. especial interest, especial value when we want to emphasise the exceptional nature of this interest or value:
- The police took especial interest in his activities and watched the house continuously.
- The Koh-i-noor diamond, now among the British crown jewels, has especial value as its history dates back to the 14th Century.
In all other cases and contexts, when it means important or different from normal, special is preferred:
- You're a very special person in my life - never forget that.
- On special occasions we have wine with our meal, but certainly not every day.
- In special cases, prisoners are allowed out on day release twice a week.
- He has such ability, I think he'll be the next special adviser to the President.
- The special effects in the Lord of the Rings films are quite mind-blowing.
- The grapes at the supermarket are on special offer - less than half price.