The words especially and specially have a slight difference. Both can be used to mean "particularly." Especially tends to be more formal, while specially tends to be more informal.

Specially can express the idea of extraordinary purpose. For example, "the procedure requires specially trained staff" or "the meal was specially prepared for our group ". Especially can express the meaning of better or more than another. For example, "I found the lesson to be especially difficult today" or "the staff at the hotel were especially helpful when we arrived late".

There is no real difference between them. The "e" part is from the old French spelling, that is all.