This is a favorite question of grammar checkers. Most of the time there is little or no difference between the words especially and specially.
Special is a common adjective. Specially is its adverb form. Special means "particular, distinguished in a distinct way, or designed for a particular purpose." Specially means "particularly, in a disintguishing manner, or for a particular purpose."
Especial is an uncommon adjective. Especially, its adverb form, is much more common. Especial means "exceptional, noteworthy, or particular." Especially means "exceptionally, in a noteworthy manner, or particularly."
In the sense of "particular" or "particularly," the words mean pretty much the same thing. Often they can be used synonymously. However, if you want to stress the exceptional or noteworthy quality, then especial or especially is a better choice. If you want to stress the distinctive purpose of something, then special or specially is the word you are looking for.
Example: This program has specially designed macros for word processors.
(A distinctive purpose)
Example: He did especially well in All-Star Game.
(A noteworthy performance)